# Distinguished waves and slopes in genus two

John Berge<EMAIL_ADDRESS>
###### Abstract.

If $R$ is a nonseparating simple closed curve on the boundary of a genus two
handlebody $H$ and the manifold $H[R]$, obtained by adding a 2-handle to $H$
along $R$, has incompressible boundary, then there exists a unique arc
$\omega$ in $\partial H$, meeting $R$ only in its endpoints, such that,
$\omega$ is isotopic in $\partial H$, keeping its endpoints on $R$, to a
nontrivial wave based at $R$ in each Heegaard diagram of $R$ on $\partial H$
which has no cut-vertex. Such a wave $\omega$ is a _distinguished-wave_ based
at $R$. Then surgery on $R$ along its distinguished wave $\omega$ yields a
pair of disjoint simple closed curves, say $m_{1}$ and $m_{2}$, in $\partial
H$, each of which represents a $\omega$-determined-slope $m$ on $\partial
H[R]$, that depends only on $R$ and $H$.

A few consequences:

  * •

Only Dehn filling of $H[R]$ at slope $m$ can yield $S^{3}$, $(S^{1}\times
S^{2})\;\\#\;L(p,q)$, or $S^{1}\times S^{2}$. So $H[R]$ embeds in at most one
of $S^{3}$, $(S^{1}\times S^{2})\;\\#\;L(p,q)$, or $S^{1}\times S^{2}$. And,
if such an embedding exists, it is unique.

  * •

Theta curves arising from unknotting tunnels of tunnel-number-one knots in
$S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$, have
canonical component knots.

  * •

One can recognize (1,1) tunnels of (1,1) knots in $S^{3}$ or $S^{1}\times
S^{2}$.

  * •

Algorithms for recognizing genus two Heegaard diagrams of $S^{3}$,
$S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$ that use waves can
be streamlined.

  * •

Efficient procedures for computing the depth of an unknotting tunnel of a knot
in $S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$ exist.

Finally, examples are provided showing that, if $R_{1}$ and $R_{2}$ are
nonseparating simple closed curves on $\partial H$ such that $H[R_{1}]$ is
homeomorphic to $H[R_{2}]$, but $(H,R_{1})$ and $(H,R_{2})$ are not
homeomorphic, then the $\omega$-determined slopes on $\partial H[R_{1}]$ and
$\partial H[R_{2}]$ may differ.

Continuing, the last section of the paper summarizes computational evidence
suggesting however that, if $\mathcal{R}$ is a set of simple closed curves on
$\partial H$ such that $R_{1}\in\mathcal{R}$ and $R_{2}\in\mathcal{R}$ means
$H[R_{1}]$ is homeomorphic to $H[R_{2}]$, then at most two distinct slopes
appear as $\omega$-determined slopes for curves in $\mathcal{R}$, and that, if
such distinct $\omega$-determined slopes exist, they are never more than
distance one apart.

CONTENTS

Section 1) Introduction.

Section 2) Preliminaries.

Section 3) Basics of genus two Heegaard diagrams, their graphs and waves based
at their curves.

Section 4) Distinguished waves are well-defined.

Section 5) Only distinguished-wave-determined slopes are meridians of tunnel-
number-one knots in $S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times
S^{2})\;\\#\;L(p,q)$.

Section 6) Distinguished waves in genus two Heegaard diagrams of closed
manifolds.

Section 7) Distinguished waves in genus two Heegaard diagrams of $S^{3}$,
$S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$.

Section 8) Some properties of meridian representatives of tunnel-number-one
knots in $S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$.

Section 9) Unknotting tunnels and canonical constituent knots of theta curves
in $S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$.

Section 10) Recognizing (1,1) tunnels of (1,1) knots in $S^{3}$ or
$S^{1}\times S^{2}$.

Section 11) Recognizing genus two Heegaard diagrams of $(S^{1}\times
S^{2})\;\\#\;L(p,q)$, $S^{3}$, or $S^{1}\times S^{2}$.

Section 12) Computing the depth of a tunnel of a tunnel-number-one knot in
$S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$.

Section 13) Other waves based at a fixed curve.

Section 14) Distinguished-wave-determined slope dependence on tunnels in tun-
nel-number-one manifolds which do not embed in $S^{3}$, $S^{1}\times S^{2}$,
or $(S^{1}\times S^{2})\;\\#\;L(p,q)$.

##  1\. Introduction

If $R$ is an essential simple closed curve in a Heegaard diagram $\mathcal{D}$
on the boundary of a handlebody $H$, and $F$ is the surface obtained by
cutting $\partial H$ open along $R$, then a _wave_ based at $R$ in
$\mathcal{D}$ is a properly embedded essential arc $w$ in $F$ such that both
endpoints of $w$ lie on the same boundary component of $F$ and $w$ is
otherwise disjoint from the curves in $\mathcal{D}$.

When $H$ has genus two, the boundary $\partial H[R]$ of the manifold $H[R]$,
obtained by adding a 2-handle to $H$ along $R$, is a torus. In this case, when
$w$ is a wave based at $R$ in a Heegaard diagram $\mathcal{D}$ of $R$ on
$\partial H$, the two boundary components, $m_{1}$ and $m_{2}$ of the regular
neighborhood of $R\cup w$ in $\partial H$, which are not isotopic to $R$ in
$\partial H$, are isotopic in $\partial H[R]$ and so specify a _wave-
determined_ _slope_ , say $m$, in $\partial H[R]$.

The results of this paper focus on such wave-determined slopes. Every
essential nonseparating simple closed curve $R$ on $\partial H$ such that
$H[R]$ has incompressible boundary has at least one wave-determined slope $w$,
and may have many. For instance, a later example shows that an infinite number
of cyclic surgery slopes of the trefoil knot are wave-determined. While most
such wave-determined slopes are ephemeral in the sense that they disappear
when the Heegaard diagram $\mathcal{D}$ of $R$ is changed, it is demonstrated
here that there always exists a _distinguished-wave_ based at $R$ such that,
$\omega$ is isotopic in $\partial H$, keeping its endpoints on $R$, to a
nontrivial wave based at $R$ in each Heegaard diagram of $R$ on $\partial H$
which has no cut-vertex. Some consequences that follow readily are:

  * •

Only Dehn filling of $H[R]$ at slope $m$ can yield $S^{3}$, $(S^{1}\times
S^{2})\;\\#\;L(p,q)$,$S^{3}$, or $S^{1}\times S^{2}$. So $H[R]$ embeds in at
most one of $S^{3}$, $(S^{1}\times S^{2})\;\\#\;L(p,q)$,$S^{3}$, or
$S^{1}\times S^{2}$. And, if such an embedding exists, it is unique.

  * •

Theta curves arising from unknotting tunnels of tunnel-number-one knots in
$S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$, have
canonical component knots.

  * •

One can recognize (1,1) tunnels of (1,1) knots in $S^{3}$ or $S^{1}\times
S^{2}$.

  * •

Algorithms for recognizing genus two Heegaard diagrams of $S^{3}$,
$S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$ that use waves can
be streamlined.

  * •

Efficient procedures for computing the depth of an unknotting tunnel of a knot
in $S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$ exist.

Finally the paper considers how distinguished-wave-determined slopes compare
in cases in which $R$ and $R^{\prime}$ are nonseparating curves on the
boundaries of genus two handlebodies $H$ and $H^{\prime}$ respectively such
that $H[R]$ and $H^{\prime}[R^{\prime}]$ are homeomorphic.

In this case, generically, the distinguished-wave-determined slope is an
invariant of the manifold $H[R]$, as results of [J06] and [ST06] show that
when $R$ is at distance greater than 5 in the curve-complex of $\partial H$
from each curve in $\partial H$ that bounds an essential disk in $H$, and
$R^{\prime}$ is a curve in the boundary of a handlebody $H^{\prime}$ of genus
two, such that $H[R]$ and $H^{\prime}[R^{\prime}]$ are homeomorphic, then the
pairs $(H,R)$ and $(H^{\prime},R^{\prime})$ are also homeomorphic.

The situation is more complicated when $H[R]$ and $H^{\prime}[R^{\prime}]$ are
homeomorphic but $(H,R)$ and $(H^{\prime},R^{\prime})$ are not homeomorphic.
To investigate how distinguished-wave-determined slopes behave in this
situation, the author’s Heegaard program 14.9 was used to look for tunnel-
number one manifolds and their distinguished-wave-determined slopes in the
subset of 59,107 one-cusped hyperbolic manifolds in SnapPy’s
‘OrientableCuspedCensus’ or ‘OCC’ of the set of all hyperbolic, orientable,
cusped manifolds which have triangulations using no more than 9 ideal
tetrahedra.

A search using Heegaard found 113,378 tunnels for 49,933 of these 59,107 one-
cusped manifolds in the OCC. So at least 49,933, or approximately 84.5% of the
one-cusped manifolds in the OCC are tunnel-number one manifolds. Discarding
the tunnel-number one manifolds which embed in $S^{3}$, $S^{1}\times S^{2}$,
or $(S^{1}\times S^{2})\,\\#\,L(p,q)$ from this set of 49,933 manifolds left a
set of 48,688 manifolds of which 7,917, or about 16.3%, have distinct
distinguished-wave-determined slopes. (Section 14 has more details on the OCC
manifolds with distinct distinguished-wave-determined slopes.)

######  Remark 1.1.

See 14.9 for links to source code for Heegaard and data files for Heegaard’s
survey of distinguished-wave-determined slopes of OCC manifolds.

##  2\. Preliminaries

Pairs of simple closed curves in a surface, say $F$, are always assumed to
have only essential intersections, up to isotopy in $F$. A properly embedded
nonseparating disk in a handlebody $H$ is a _meridian disk_ of $H$. A set
$\Delta$ of pairwise disjoint meridian disks in a handlebody $H$ is a
_complete set_ of meridian disks of $H$ if cutting $H$ open along the disks in
$\Delta$ yields a 3-ball. Then the _genus_ of $H$ is equal to the number of
disks in $\Delta$, which in turn is equal to the genus of the closed connected
orientable surface $\partial H$.

A finite, nonempty set $\mathcal{C}$ of pairwise disjoint essential simple
closed curves in the boundary of a handlebody $H$ together with a set
$\partial\Delta$ of simple closed curves in $\partial H$ which bound the disks
of a complete set of meridian disks $\Delta$ of $H$ is a _Heegaard diagram_
$\mathcal{D}$. (We usually assume there is no curve $c\in\mathcal{C}$ that
bounds a disk in $H$. And, since $\mathcal{D}$ is generally nonplanar, usually
work with the planar diagram $\mathcal{D}^{\prime}$ obtained by cutting $H$
open along the disks in $\Delta$.)

The operation of cutting $H$ open along the disks in $\Delta$ cuts $H$ into a
3-ball $\mathcal{B}$ with two copies of each disk of $\Delta$ in
$\partial\mathcal{B}$. These disks in $\partial\mathcal{B}$ form the
‘fat’-vertices of a planar graph $G(\mathcal{D})$ whose edges are the arcs
obtained by cutting each curve $c\in\mathcal{C}$ open at the points in
$c\cap\partial\Delta$. Then $\mathcal{D}$ and $\mathcal{D}^{\prime}$ can
recovered from $G(\mathcal{D})$ together with some additional information
which indicates how each pair of ‘fat’-vertices of $G$ arising from cutting
$H$ open along a disk in $\Delta$, must be identified to reconstitute $H$ and
the simple closed curves in $\mathcal{C}$.

The _complexity_ of the Heegaard diagram of $\mathcal{C}$ and $\partial\Delta$
in $\partial H$ is the total number of essential intersections of
$\mathcal{C}$ and $\partial\Delta$ in $\partial H$.

If $c$ is a simple closed curve and $\mathcal{D}$ is a Heegaard diagram to
which $c$ contributes edges, let $|c|$ denote the _complexity_ of $c$ in
$\mathcal{D}$, which we take to be equal to the total number of edges which
$c$ contributes to $\mathcal{D}$. More generally, we use $|X|$ to denote the
number of elements of a finite set $X$.

The graph terminology we use is standard and basic.

If $v$ is a vertex of a connected graph $G$ such that deleting $v$ and the
interiors of edges of $G$ meeting $v$ from $G$ disconnects $G$, we say $v$ is
a _cut-vertex_ of $G$.

An edge $e$ of a graph $G$ is a _loop_ in $G$ if there is a vertex $v$ of $G$
containing both ends of $e$.

An element $x$ of a free group $F$ is a _primitive_ element of $F$ if $x$ is a
free generator of $F$. If $H$ is a handlebody of genus $g$, then $\pi_{1}(H)$
is a free group of rank $g$, and it is well-known that if $c$ is a simple
closed curve in $\partial H$, then $c$ is primitive in $\pi_{1}(H)$ if and
only if there is a meridian disk of $H$ whose boundary intersects $c$
transversely in a single point.

Let $\Sigma$ be a closed orientable surface, and let $\mathcal{C}$ be a finite
set of simple closed curves in $\Sigma$. Let $c$ be a member of $\mathcal{C}$
and let $F$ be the surface with two boundary components $c^{+}$, $c^{-}$
obtained by cutting $\Sigma$ open along $c$. Then a _wave based_ at $c$ is a
properly embedded arc $\omega$ in $F$ such that the interior of $\omega$ is
disjoint from each curve in $\mathcal{C}$, both endpoints of $\omega$ lie on
the same boundary component of $F$, and $\omega$ is essential in $F$. (If we
need to be specific about the _side_ of $c$ at which $\omega$ is based, we
speak of $\omega$ as _based at_ $c^{+}$, respectively $c^{-}$, depending upon
whether the endpoints of $\omega$ lie on $c^{+}$, respectively $c^{-}$.)

######  Remark 2.1.

Note that because of the symmetry of genus two Heegaard diagrams induced by
the omnipresent hyperelliptic involution of genus two Heegaard surfaces
[HS89], waves based at nonseparating simple closed curves in genus two
Heegaard diagrams always arise in pairs whose members are exchanged by the
action of the hyperelliptic involution, with one member of each pair based at
the ‘+’ side of a curve and the other member of the pair based at the ‘-’ side
of a curve, as in Figure 2b, for example.

With this understood, we will often speak of a wave as being ‘unique’ when we
should perhaps say something like “the pair of waves equivalent under the
action of the hyperelliptic involution is unique up to proper isotopy in the
Heegaard surface …”.

######  Remark 2.2.

Figures in this paper will often display only one member of each pair of waves
related by the hyperelliptic involution.

######  Definition 2.3.

Wave-determined slope representatives.

Suppose $\mathcal{D}$ is a Heegaard diagram of a nonseparating simple closed
curve $R$ on the boundary of a genus two handlebody $H$, $\omega$ is a wave to
$R$ in $\mathcal{D}$, and $N$ is a regular neighborhood of $R\cup\omega$ in
$\partial H$. Then $N$ is a pair of pants, and we say the two boundary
components of $N$, not isotopic to $R$ in $\partial H$, say $m_{1}$ and
$m_{2}$, are _wave-determined slope representatives_.

######  Definition 2.4.

Wave-determined slopes.

While wave-determined slope representatives are not isotopic in $\partial H$,
they are isotopic unoriented curves in the torus boundary of $H[R]$, and so
represent a _wave-determined slope_ $s$ on $\partial H[R]$.

######  Definition 2.5.

Surgery along a wave.

Suppose $R$ is a nonseparating simple closed curve in an orientable surface
$\Sigma$ and $\omega$ is a wave in $\Sigma$ based at $R$. Then a regular
neighborhood $N$ of $R\cup\omega$ in $\Sigma$ is a pair of pants with three
simple closed curve boundary components, one of which is isotopic to $R$. If
$m$ is one of the other boundary components of $N$, we say $m$ was obtained by
_surgery along $R$_.

##  3\. Basics of genus two Heegaard diagrams, their graphs and waves based at
their curves

######  Lemma 3.1.

_Graphs of genus two Heegaard diagrams._

If $\mathcal{C}$ is a finite set of disjoint essential simple closed curves on
the boundary of a genus two handlebody $H$, $\\{D_{A},D_{B}\\}$ is a complete
set of meridian disks for $H$, and no curve in $\mathcal{C}$ is disjoint from
$\partial D_{A}\cup\partial D_{B}$, then the graph of the Heegaard diagram of
the curves in $\mathcal{C}$ with respect to $\\{D_{A},D_{B}\\}$ has the form
of one of the three graphs in Figure  _1_.

###### Proof.

This is a result of [O79]. ∎

Figure 1. Three types of graphs of genus two Heegaard diagrams. Here
$a,b,c,d\geq 0$ indicate how many parallel edges connect each pair of
vertices.

######  Lemma 3.2.

_Reducing the complexity of Heegaard diagrams with cut-vertices._

If a Heegaard diagram $\mathcal{D}$ of a set of pairwise disjoint essential
simple closed curves in the boundary of a genus two handlebody $H$ has a cut-
vertex, $\mathcal{D}$ has the form of Figure _1c_ or Figure  _2a_. Then there
are waves based at $\partial D_{A}^{+}$ and $\partial D_{A}^{-}$ in
$\mathcal{D}$, as shown in Figure  _2b_, and, if $a+b>0$ in Figure _2a_, the
complexity of $\mathcal{D}$ can be reduced by an amount equal to $a+2b$ by
replacing the meridional disk $D_{A}$ of $H$ with the meridional disk $D_{C}$
of $H$ bounded by the curve $C$ in Figure  _2c_.

###### Proof.

Examine Figure 2. ∎

Figure 2. Reducing the complexity of Heegaard diagrams with cut-vertices.

###  3.1. Finding minimal complexity genus two Heegaard diagrams

Although the following two lemmas are certainly well-known, because of the
central roles they play here, we outline proofs.

######  Lemma 3.3.

Suppose $\mathcal{C}$ is a finite set of pairwise disjoint simple closed
curves on the boundary of a genus two handlebody $H$, and $\Delta$ is a
complete set of meridian disks for $H$.

If there exists a complete set of meridian disks $\Delta^{\prime}$ for $H$
such that
$|\partial\Delta\cap\mathcal{C}|>|\partial\Delta^{\prime}\cap\mathcal{C}|$,
then there also exists a complete set of meridian disks $\Delta^{*}$ for $H$
such that
$|\partial\Delta\cap\mathcal{C}|>|\partial\Delta^{*}\cap\mathcal{C}|$, and
$\Delta^{*}$ is obtained from $\Delta$ by replacing one disk of $\Delta$ with
a disk in $H$ whose boundary is a bandsum in $\partial H$ of the boundaries of
the two disks in $\Delta$.

It follows that there exists a finite sequence $\\{\Delta_{i}\\}$, $0\leq
i\leq n$, of complete sets of meridian disks of $H$ such that:
$\Delta_{0}=\Delta$; $\partial\Delta_{n}\cap\mathcal{C}$ is minimal; for each
$i$ with $0\leq i<n$,
$|\partial\Delta_{i}\cap\mathcal{C}|>|\partial\Delta_{i+1}\cap\mathcal{C}|$;
and, for each $i$ with $0\leq i<n$, $\Delta_{i+i}$ is obtained from
$\Delta_{i}$ by replacing one of the disks of $\Delta_{i}$ with a disk whose
boundary is a bandsum in $\partial H$ of the boundaries of the two disks in
$\Delta_{i}$.

###### Proof.

Given $\mathcal{C}$ on $\partial H$, let $min(\mathcal{C})$ consist of the
complete sets of meridian disks of $H$ which intersect the curves in
$\mathcal{C}$ minimally. And then choose $\Delta^{\prime}\in min(\mathcal{C})$
so that $|\Delta^{\prime}\cap\Delta|$ consists of a minimal number of arcs.
(Recall that because handlebodies are aspherical, simple closed curves of
intersection of two disks in $H$ can be removed. So we may assume disks in
$\Delta$ and $\Delta^{\prime}$ intersect only in arcs.) Then there are two
possibilities.

If $\Delta$ and $\Delta^{\prime}$ are disjoint, then either $\Delta$ and
$\Delta^{\prime}$ are isotopic in $H$, or $\Delta^{\prime}$ is obtained from
$\Delta$ by replacing one disk of $\Delta$ with a disk in $H$ whose boundary
is a bandsum in $\partial H$ of the boundaries of the two disks in $\Delta$.
In which case, we may take $\Delta^{*}$ to be $\Delta^{\prime}$.

If $\Delta$ and $\Delta^{\prime}$ are not disjoint, then there exist outermost
subdisks of both $\Delta$ and $\Delta^{\prime}$ that are cut off of disks of
$\Delta$ and $\Delta^{\prime}$ by arcs of intersection in
$\Delta\cap\Delta^{\prime}$. In this case, choose $D$ among the outermost
subdisks of $\Delta$ so that $|D\cap\mathcal{C}|$ is minimal, and set
$m=|D\cap\mathcal{C}|$. Similarly, choose $D^{\prime}$ among the outermost
subdisks of $\Delta^{\prime}$ so that $|D^{\prime}\cap\mathcal{C}|$ is
minimal, and set $m^{\prime}=|D^{\prime}\cap\mathcal{C}|$.

Now suppose $\Delta=\\{D_{A},D_{B}\\}$. Then, since $D^{\prime}$ is outermost,
$D^{\prime}$ meets one member of $\\{D_{A},D_{B}\\}$ in a single arc of
intersection, say $\alpha$, lying in $\partial D^{\prime}$, and $D^{\prime}$
is disjoint from the other member of $\\{D_{A},D_{B}\\}$. Supposing $\alpha$
lies in, say $D_{A}$, $\alpha$ cuts $D_{A}$ into two subdisks, say $D_{A_{1}}$
and $D_{A_{2}}$, each of which contains an outermost subdisk meeting
$\mathcal{C}$ at least $m$ times.

Next, note both $D_{A_{1}}\cup D_{m^{\prime}}$ and $D_{A_{2}}\cup
D_{m^{\prime}}$ are nonseparating disks in $H$, with one member of
$\\{D_{A_{1}}\cup D_{m^{\prime}},D_{A_{2}}\cup D_{m^{\prime}}\\}$ isotopic to
$D_{B}$ in $H$, while the boundary of the other member of $\\{D_{A_{1}}\cup
D_{m^{\prime}},D_{A_{2}}\cup D_{m^{\prime}}\\}$ is a bandsum in $\partial H$
of $\partial D_{A}$ and $\partial D_{B}$. In which case, the other member of
$\\{D_{A_{1}}\cup D_{m^{\prime}},D_{A_{2}}\cup D_{m^{\prime}}\\}$ together
with $D_{B}$ forms a complete set of meridian disks for $H$.

Supposing for the moment that $m>m^{\prime}$, the replacement of $D_{A}$ with
the member of $\\{D_{A_{1}}\cup D_{m^{\prime}},D_{A_{2}}\cup
D_{m^{\prime}}\\}$, not isotopic to $D_{B}$ in $H$, yields a new complete set
of meridian disks for $H$, which can be taken as the complete set of meridian
disks $\Delta^{*}$ of the hypothesis.

It is possible to see that $m>m^{\prime}$ holds by observing that the
alternative $m\leq m^{\prime}$ is not compatible with the choice of
$\Delta^{\prime}$. For, if $m\leq m^{\prime}$ held, the outermost disk $D$ of
$\Delta$ could be used to perform surgery on a member of $\Delta^{\prime}$,
demonstrating that either $\Delta^{\prime}$ does not lie in
$min(\mathcal{C})$, or $\Delta^{\prime}$ lies in $min(\mathcal{C})$, but there
exists another complete set of meridian disks for $H$, say $\Delta^{\prime*}$,
in $min(\mathcal{C})$ such that
$|\Delta\cap\Delta^{\prime*}|<|\Delta\cap\Delta^{\prime}|$.

Finally, it should be clear that the procedure above can be performed
repeatedly and will produce a sequence of complete sets of meridian disks for
$H$ with the properties claimed in the statement of the lemma. ∎

######  Lemma 3.4.

Suppose $\mathcal{C}$ is a finite set of pairwise disjoint simple closed
curves on the boundary of a genus two handlebody $H$, and $\Delta$ is a
complete set of meridian disks for $H$ such that no complete set $\Delta^{*}$
of meridian disks for $H$ obtained by replacing one member of $\Delta$ with a
disk in $H$ whose boundary is a bandsum in $\partial H$ of the boundaries of
the two disks in $\Delta$ satisfies
$|\partial\Delta\cap\mathcal{C}|>|\partial\Delta^{*}\cap\mathcal{C}|$. If
$\Delta^{\prime}$ is also a complete set of meridian disks for $H$ such that
no complete set $\Delta^{\prime*}$ of meridian disks for $H$ obtained by
replacing one member of $\Delta^{\prime}$ with a disk in $H$ whose boundary is
a bandsum in $\partial H$ of the boundaries of the two disks in
$\Delta^{\prime}$ satisfies
$|\partial\Delta^{\prime}\cap\mathcal{C}|>|\partial\Delta^{\prime*}\cap\mathcal{C}|$,
then there exists a finite sequence $\\{\Delta_{i}\\}$, $0\leq i\leq n$, of
complete sets of meridian disks of $H$ such that: $\Delta_{0}=\Delta$;
$\Delta_{n}=\Delta^{\prime}$; for each $i$ with $0\leq i<n$,
$|\partial\Delta\cap\mathcal{C}|=|\partial\Delta^{\prime}\cap\mathcal{C}|$;
and, for each $i$ with $0\leq i<n$, $\Delta_{i+i}$ is obtained from
$\Delta_{i}$ by replacing one of the disks of $\Delta_{i}$ with a disk whose
boundary is a bandsum in $\partial H$ of the boundaries of the two disks in
$\Delta_{i}$. In particular,
$|\partial\Delta\cap\mathcal{C}|=|\partial\Delta^{\prime}\cap\mathcal{C}|$.

###### Proof.

The argument is similar to that above, and again there are two cases.

If $\Delta$ and $\Delta^{\prime}$ are disjoint in $H$, then either $\Delta$
and $\Delta^{\prime}$ are isotopic in $H$, or $\Delta^{\prime}$ is obtained
from $\Delta$ by replacing one of the disks of $\Delta$ with a disk whose
boundary in $\partial H$ is a bandsum of the boundaries of the disks in
$\Delta$, and similarly, $\Delta$ is obtained from $\Delta^{\prime}$ by
replacing one of the disks of $\Delta^{\prime}$ with a disk whose boundary in
$\partial H$ is a bandsum of the boundaries of the disks in $\Delta^{\prime}$.
Then the hypothesis have ruled out the possibility that
$|\partial\Delta\cap\mathcal{C}|<|\partial\Delta^{\prime}\cap\mathcal{C}|$ or
that
$|\partial\Delta^{\prime}\cap\mathcal{C}|<|\partial\Delta\cap\mathcal{C}|$. So
$|\partial\Delta\cap\mathcal{C}|=|\partial\Delta^{\prime}\cap\mathcal{C}|$.

If $\Delta$ and $\Delta^{\prime}$ are not disjoint, then there exist outermost
subdisks of both $\Delta$ and $\Delta^{\prime}$ that are cut off of disks of
$\Delta$ and $\Delta^{\prime}$ by arcs of intersection in
$\Delta\cap\Delta^{\prime}$. In this case, choose $D$ among the outermost
subdisks of $\Delta$ so that $|D\cap\mathcal{C}|$ is minimal, and set
$m=|D\cap\mathcal{C}|$. Similarly, choose $D^{\prime}$ among the outermost
subdisks of $\Delta^{\prime}$ so that $|D^{\prime}\cap\mathcal{C}|$ is
minimal, and set $m^{\prime}=|D^{\prime}\cap\mathcal{C}|$.

Now suppose $\Delta=\\{D_{A},D_{B}\\}$. Then, since $D^{\prime}$ is outermost,
$D^{\prime}$ meets one member of $\\{D_{A},D_{B}\\}$ in a single arc of
intersection, say $\alpha$, lying in $\partial D^{\prime}$, and $D^{\prime}$
is disjoint from the other member of $\\{D_{A},D_{B}\\}$. Supposing $\alpha$
lies in, say $D_{A}$, $\alpha$ cuts $D_{A}$ into two subdisks, say $D_{A_{1}}$
and $D_{A_{2}}$, each of which contains an outermost subdisk meeting
$\mathcal{C}$ at least $m$ times.

Next, note both $D_{A_{1}}\cup D_{m^{\prime}}$ and $D_{A_{2}}\cup
D_{m^{\prime}}$ are nonseparating disks in $H$, with one member of
$\\{D_{A_{1}}\cup D_{m^{\prime}},D_{A_{2}}\cup D_{m^{\prime}}\\}$ isotopic to
$D_{B}$ in $H$, while the boundary of the other member of $\\{D_{A_{1}}\cup
D_{m^{\prime}},D_{A_{2}}\cup D_{m^{\prime}}\\}$ is a bandsum in $\partial H$
of $\partial D_{A}$ and $\partial D_{B}$. In which case, the other member of
$\\{D_{A_{1}}\cup D_{m^{\prime}},D_{A_{2}}\cup D_{m^{\prime}}\\}$ together
with $D_{B}$ forms a complete set of meridian disks $\Delta^{*}$ for $H$.

Recalling that by hypothesis,
$|\Delta^{*}\cap\mathcal{C}|\geq|\Delta\cap\mathcal{C}|$, we see $m\leq
m^{\prime}$.

A similar argument, replacing $D^{\prime}$ with $D$ and $\Delta$ with
$\Delta^{\prime}$, shows that $D$ can be used to produce a complete set
$\Delta^{\prime*}$ of meridian disks for $H$ which is obtained from
$\Delta^{\prime}$ by replacing one disk of $\Delta^{\prime}$ with a disk whose
boundary is a bandsum in $\partial H$ of the boundaries of the two disks in
$\Delta^{\prime}$. However, by hypothesis,
$|\Delta^{\prime*}\cap\mathcal{C}|\geq|\Delta^{\prime}\cap\mathcal{C}|$, which
implies $m^{\prime}\leq m$.

So $m=m^{\prime}$, $|\Delta^{*}\cap\mathcal{C}|=|\Delta\cap\mathcal{C}|$ and
$|\Delta^{\prime*}\cap\mathcal{C}|=|\Delta^{\prime}\cap\mathcal{C}|$. In
addition, since $D^{\prime}$ is a subdisk of $\Delta^{\prime}$,
$|\Delta\cap\Delta^{\prime}|>|\Delta^{*}\cap\Delta^{\prime}|$. And similarly,
since $D$ is a subdisk of $\Delta$,
$|\Delta\cap\Delta^{\prime}|>|\Delta^{\prime*}\cap\Delta|$.

It follows that the desired finite sequence $\\{\Delta_{i}\\}$, $0\leq i\leq
n$, of complete sets of meridian disks of $H$ exists such that:
$\Delta_{0}=\Delta$; $\Delta_{n}=\Delta^{\prime}$; for each $i$ with $0\leq
i<n$,
$|\partial\Delta\cap\mathcal{C}|=|\partial\Delta^{\prime}\cap\mathcal{C}|$;
and, for each $i$ with $0\leq i<n$, $\Delta_{i+i}$ is obtained from
$\Delta_{i}$ by replacing one of the disks of $\Delta_{i}$ with a disk whose
boundary is a bandsum in $\partial H$ of the boundaries of the two disks in
$\Delta_{i}$. ∎

###  3.2. Three types of bandsums in genus two

Suppose $\Delta$ = $\\{D_{A},D_{B}\\}$ is a complete set of meridian disks for
$H$ comprised of meridian disks $D_{A}$ and $D_{B}$ for $H$, and $\mathcal{D}$
is the Heegaard diagram of the curves in $\mathcal{C}$ with respect to
$\partial\Delta$.

Consider the effect on $\mathcal{D}$ of replacing one of $D_{A},D_{B}$ with a
meridian disk $D_{C}$ of $H$ whose boundary $C$ in $\partial H$ is a bandsum
of $\partial D_{A}$ with $\partial D_{B}$. This replaces $\mathcal{D}$ with a
new Heegaard diagram $\mathcal{D}^{\prime}$ of the curves in $\mathcal{C}$
with respect to the new set of complete meridian disks for $H$. The change in
diagrams in passing from $\mathcal{D}$ to $\mathcal{D}^{\prime}$ is determined
by the band $\beta$ used to band $\partial D_{A}$ and $\partial D_{B}$
together to form $\partial D_{C}$. Figure 3 shows three relevant
configurations of $\beta$ with respect to the edges and faces of
$\mathcal{D}$. Namely, configurations in which:

  1. (1)

$\beta$ has essential intersections with edges of $\mathcal{D}$.

  2. (2)

$\beta$ lies in a face of $\mathcal{D}$, but $\beta$ is not properly isotopic
to an edge of $\mathcal{D}$.

  3. (3)

$\beta$ lies in a face of $\mathcal{D}$ and $\beta$ is properly isotopic to an
edge of $\mathcal{D}$.

Then examination of Figure 3 provides some details about the changes that
occur to $\mathcal{D}$ in passing from $\mathcal{D}$ to $\mathcal{D}^{\prime}$
in each of the three cases enumerated above. (Here let $|\\{D_{A},D_{B}\\}|$
denote the complexity of $\mathcal{D}$, and let $|\\{D_{A},D_{C}\\}|$,
respectively $|\\{D_{B},D_{C}\\}|$, denote the complexity of
$\mathcal{D}^{\prime}$ when $\mathcal{D}^{\prime}$ uses compete sets of
meridian disks $|\\{D_{A},D_{C}\\}|$, respectively $|\\{D_{B},D_{C}\\}|$ of
$H$.)

  1. (1)

$\beta$ has essential intersections with edges of $\mathcal{D}$.
In this case, $\mathcal{D}^{\prime}$ has a cut-vertex, since there is a wave
$\omega$ to $C$ in $\mathcal{D}^{\prime}$. Also $|\\{D_{A},D_{C}\\}|$ \-
$|\\{D_{A},D_{B}\\}|>a>0$, and $|\\{D_{B},D_{C}\\}|$ \-
$|\\{D_{A},D_{B}\\}|>b>0$. So the complexity of $\mathcal{D}^{\prime}$ is
always greater than the complexity of $\mathcal{D}$ in this case.

  2. (2)

$\beta$ lies in a face of $\mathcal{D}$ and $\beta$ is not properly isotopic
to an edge of $\mathcal{D}$.
In this case, $\mathcal{D}^{\prime}$ has a cut-vertex, since there is a wave
$\omega$ to $C$ in $\mathcal{D}^{\prime}$. Also $|\\{D_{A},D_{C}\\}|$ \-
$|\\{D_{A},D_{B}\\}|$ = $a>0$, and $|\\{D_{B},D_{C}\\}|$ \-
$|\\{D_{A},D_{B}\\}|$ = $b>0$. So the complexity of $\mathcal{D}^{\prime}$ is
again always greater than the complexity of $\mathcal{D}$ in this case.

  3. (3)

$\beta$ lies in a face of $\mathcal{D}$ and $\beta$ is properly isotopic to an
edge of $\mathcal{D}$.
In this case, Lemmas 3.8 and 3.9 guarantee that $\mathcal{D}^{\prime}$ is
connected and has no cut-vertex. Then $|\\{D_{A},D_{C}\\}|$ \-
$|\\{D_{A},D_{B}\\}|$ = $a-c$, and $|\\{D_{B},D_{C}\\}|$ \-
$|\\{D_{A},D_{B}\\}|$ = $b-c$. So, depending on the relative magnitudes of
$a,b$, and $c$, and whether $D_{C}$ replaces $D_{A}$ or $D_{B}$, the
complexity of $\mathcal{D}^{\prime}$ may be less than, equal to, or greater
than that of $\mathcal{D}$.

In any case, we see the following lemmas hold:

######  Lemma 3.5.

_Only bandsums along edges of Heegaard diagrams do not necessarily increase
complexity._

######  Lemma 3.6.

_Only bandsums along edges of Heegaard diagrams do not necessarily introduce
cut-vertices._

Figure 3. Three types of bandsums. Here $a,b,c>0$, and boundary components $A$
and $B$ of each shaded pair of pants should be thought of as connected by a
band $\beta$ disjoint from the curve $C$.

######  Definition 3.7.

Positive and Nonpositive connected diagrams.

Say a connected Heegaard diagram is _positive_ if the curves of the diagram
can be oriented so that all intersections of curves in the diagram are
positive. Otherwise, say the diagram is _nonpositive_.

######  Lemma 3.8.

_Bandsums along edges of nonpositive diagrams yield connected diagrams without
cut-vertices._

Suppose $H$ is a genus two handlebody with a complete set of meridian disks
$\\{D_{A},D_{B}\\}$, and $\mathcal{D}$ is a connected nonpositive Heegaard
diagram, without cut-vertices, of a nonseparating simple closed curve $R$ on
$\partial H$ with respect to $\\{D_{A},D_{B}\\}$. If one of $D_{A},D_{B}$, say
$D_{B}$, is replaced by a meridian disk $D_{C}$ in $H$ whose boundary $C$ is a
bandsum of $\partial D_{A}$ and $\partial D_{B}$ along an edge of
$\mathcal{D}$, then the resulting Heegaard diagram $\mathcal{D}^{\prime}$ of
$R$ with respect to $\\{D_{A},D_{C}\\}$ is connected and has no cut-vertices.

###### Proof.

Consider Figure 4, which displays the relevant subdiagram of $\mathcal{D}$
together with the curve $C$. Observe that $\mathcal{D}^{\prime}$ will be
connected and have no cut-vertices if each vertex of $\mathcal{D}^{\prime}$ is
connected to at least two other vertices in $\mathcal{D}^{\prime}$.

To see this is the case, consider, for example, vertex $A^{+}$ of
$\mathcal{D}^{\prime}$. Since $b>0$, there is an edge $e$ of
$\mathcal{D}^{\prime}$ connecting $A^{+}$ to vertex $C^{+}$ of
$\mathcal{D}^{\prime}$. And, since $a>0$, there is an edge $f$ of
$\mathcal{D}^{\prime}$ which is the union of an edge of $\mathcal{D}$
connecting $A^{+}$ to $B^{-}$ and an edge of $\mathcal{D}$ connecting $B^{+}$
to one of $A^{-}$ or $C^{-}$. So $A^{+}$ is connected to at least two distinct
vertices in $\mathcal{D}^{\prime}$.

Similarly, one verifies that each of the other vertices of
$\mathcal{D}^{\prime}$ is connected to at least two vertices of
$\mathcal{D}^{\prime}$. It follows $\mathcal{D}^{\prime}$ is connected and has
no cut-vertices. ∎

Figure 4. Bandsums along edges of nonpositive diagrams yield connected
diagrams without cut-vertices. Here $a>0$ and $b>0$.

######  Lemma 3.9.

_Bandsums Along Edges of Positive Diagrams Yield Connected, Nonpositive
Diagrams Without Cut-Vertices._

Suppose $H$ is a genus two handlebody with a complete set of meridian disks
$\\{D_{A},D_{B}\\}$, and $\mathcal{D}$ is a connected positive Heegaard
diagram, without cut-vertices, of a nonseparating simple closed curve $R$ on
$\partial H$ with respect to $\\{D_{A},D_{B}\\}$. If one of $D_{A},D_{B}$, say
$D_{B}$, is replaced by a meridian disk $D_{C}$ in $H$ whose boundary $C$ is a
bandsum of $\partial D_{A}$ and $\partial D_{B}$ along an edge of
$\mathcal{D}$, then the resulting Heegaard diagram $\mathcal{D}^{\prime}$ of
$R$ with respect to $\\{D_{A},D_{C}\\}$ is connected, nonpositive, and has no
cut-vertices.

###### Proof.

Figure 5 displays $\mathcal{D}$, the curve $C$, and a horizontal wave
$\omega_{h}$ based at $R$. First, observe that $c>0$ and $d>0$ in Figure 5
imply $C$ has both positive and negative signed intersections with $R$. So
$\mathcal{D}^{\prime}$ is nonpositive. Next, note it is enough to show that
each vertex of $\mathcal{D}^{\prime}$ is connected to at least two other
vertices of $\mathcal{D}^{\prime}$ in order to conclude $\mathcal{D}^{\prime}$
is connected and has no cut-vertices.

Consider, for example, vertex $A^{+}$ of $\mathcal{D}^{\prime}$. Since $a>0$,
there is an edge $f$ of $\mathcal{D}^{\prime}$ which is the union of an edge
of $\mathcal{D}$ connecting $A^{+}$ to $B^{-}$ and an edge of $\mathcal{D}$
connecting $B^{+}$ to one of $A^{-}$ or $C^{-}$. In addition, since $c>0$,
$A^{+}$ is connected to $C^{+}$ in $\mathcal{D}^{\prime}$. So $A^{+}$ is
connected to at least two distinct vertices in $\mathcal{D}^{\prime}$.

Similarly, one verifies that each of the other vertices of
$\mathcal{D}^{\prime}$ is connected to at least two other vertices of
$\mathcal{D}^{\prime}$. It follows $\mathcal{D}^{\prime}$ is connected and has
no cut-vertices. ∎

Figure 5. Bandsums along edges of positive diagrams yield connected diagrams
without cut-vertices. Here $a,c,d>0$.

###  3.3. Waves based at nonseparating simple closed curves in connected genus
two diagrams without cut-vertices

######  Lemma 3.10.

_A nonseparating curve in a nonpositive diagram has unique waves._

Suppose H is a genus two handlebody, $\\{D_{A},D_{B}\\}$ is a complete set of
meridian disks for H and R is a nonseparating simple closed curve on $\partial
H$ such that the Heegaard diagram $\mathcal{D}$ of R with respect to
$\\{D_{A},D_{B}\\}$ is nonpositive. Then there is a unique wave based at
$R^{+}$, and a unique wave based at $R^{-}$, in $\mathcal{D}$.

###### Proof.

Consider Figure 6. ∎

Figure 6. Genus two nonpositive Heegaard diagrams of nonseparating simple
closed curves have unique waves. Figure 6 shows a twice-punctured torus $F$
obtained by cutting the boundary of a genus two handlebody $H$ open along a
nonseparating simple closed curve $R$ in $\partial H$. If $\\{D_{A},D_{B}\\}$
is a complete set of meridian disks of $H$ such that the Heegaard diagram
$\mathcal{D}$ of $R$ with respect to $\\{D_{A},D_{B}\\}$ is nonpositive, there
is a properly embedded subarc $b^{+}$ of $\partial D_{A}\cup\partial D_{B}$ in
$F$ with its endpoints on $R^{+}$, and a properly embedded subarc $b^{-}$ of
$\partial D_{A}\cup\partial D_{B}$ in $F$ with its endpoints on $R^{-}$. Then
it is easy to see that any wave $\omega^{+}$ based at $R^{+}$ in $\mathcal{D}$
is properly isotopic in $F$ to $b^{+}$. Similarly, any wave $\omega^{-}$ based
at $R^{-}$ in $\mathcal{D}$ is properly isotopic in $F$ to $b^{-}$.

######  Remark 3.11.

Although nonpositive genus two Heegaard diagrams with cut-vertices have unique
waves, the waves obtained from different nonpositive diagrams may be distinct.
Example 13.9 shows there exist an infinite number of distinct nonpositive
genus two Heegaard diagrams of a trefoil relator $R$ such that waves based at
$R$ in distinct nonpositive diagrams are also distinct.

######  Definition 3.12.

Horizontal and Vertical waves in positive diagrams without cut-vertices.

Suppose $\mathcal{D}$ is a positive, connected genus two Heegaard diagram,
without cut-vertices, of a nonseparating simple closed curve. Then
$\mathcal{D}$ has the form of Figure 7a with vertices $A^{+}$, $A^{-}$,
$B^{+}$, and $B^{-}$, and weights $a,c,d>0$.

A wave $\omega$ in $\mathcal{D}$ is a _vertical_ wave if one endpoint of
$\omega$ lies on an edge of $\mathcal{D}$ connecting $A^{+}$ to one member of
$\\{B^{+},B^{-}\\}$, while the other endpoint of $\omega$ lies on an edge of
$\mathcal{D}$ connecting $A^{-}$ to the other member of $\\{B^{+},B^{-}\\}$.

A wave $\omega$ in $\mathcal{D}$ is a _horizontal_ wave if one endpoint of
$\omega$ lies on an edge of $\mathcal{D}$ connecting vertices $A^{+}$ and
$A^{-}$, while the other endpoint of $\omega$ lies on an edge of $\mathcal{D}$
connecting vertices $B^{+}$ and $B^{-}$.

Figure 7. Horizontal and vertical waves in positive diagrams without cut-
vertices. A Heegaard diagram $\mathcal{D}$ of a nonseparating simple closed
curve $R$ on the boundary of a genus two handlebody $H$ is a _positive
diagram_ if the graph of $\mathcal{D}$ has the form of Figure 7a with
$a,c,d>0$. Then $\mathcal{D}$ has two waves $\omega_{h}$ and $\omega_{v}$, as
shown in Figure 7b. In particular, $\mathcal{D}$ has a _horizontal wave_
$\omega_{h}$ with one endpoint of $\omega_{h}$ on an edge of Figure 7b
connecting vertices $A^{-}$ to $A^{+}$, and the other endpoint of $\omega_{h}$
on an edge of Figure 7b connecting vertices $B^{-}$ to $B^{+}$. In addition,
$\mathcal{D}$ has a _vertical wave_ $\omega_{v}$ with one endpoint of
$\omega_{v}$ on an edge of Figure 7b connecting vertices $B^{-}$ to $A^{+}$,
and the other endpoint of $\omega_{v}$ on an edge of Figure 7b connecting
vertices $A^{-}$ to $B^{+}$.

######  Definition 3.13.

The distinguished wave at a nonseparating curve.

Suppose $R$ is a nonseparating simple closed curve on the boundary of a genus
two handlebody $H$, $H[R]$ has incompressible boundary, and
$\\{D_{A},D_{B}\\}$ is a complete set of meridian disks for $H$ such that the
Heegaard diagram $\mathcal{D}$ of $R$ with respect to $\\{D_{A},D_{B}\\}$ is
connected, and has no cut-vertex.

  1. (1)

If $\mathcal{D}$ is nonpositive, let $\omega$ be the unique wave based at $R$
in $\mathcal{D}$.

  2. (2)

If $\mathcal{D}$ is positive, let $\omega$ be the horizontal wave $\omega_{h}$
based at $R$ in $\mathcal{D}$.

Then $\omega$ is the _distinguished wave_ based at $R$ in $\partial H$.

##  4\. Distinguished waves are well-defined.

######  Theorem 4.1.

_Distinguished waves are well-defined._

Suppose $R$ is a nonseparating simple closed curve on the boundary of a genus
two handlebody $H$, $\Delta$ is a complete set of meridian disks for $H$, the
Heegaard diagram $\mathcal{D}$, of $R$ with respect to $\Delta$ is connected
and has no cut-vertex, and $\omega$ is a distinguished wave based at $R$ in
$\mathcal{D}$. Then the distinguished wave $\omega$ based at $R$ is well-
defined. In other words, $\omega$ depends only on the embedding of $R$ in
$\partial H$.

In particular, if $\Delta^{\prime}$ is another complete set of meridian disks
for $H$ such that the Heegaard diagram $\mathcal{D}^{\prime}$ of $R$ with
respect to $\Delta^{\prime}$ is connected and has no cut-vertex, then $\omega$
also appears as a distinguished wave based at $R$ in $\mathcal{D}^{\prime}$,
up to proper isotopy of $\omega$ in $\partial H$ keeping the endpoints of
$\omega$ on $R$.

###### Proof.

Since both $\mathcal{D}$ and $\mathcal{D}^{\prime}$ are connected and have no
cut-vertex, Lemma 4.2 implies there exist finite sequences $\\{\Delta_{i}\\}$,
$0\leq i\leq n$, of complete sets of meridian disks for $H$, and Heegaard
diagrams $\\{\mathcal{D}_{i}\\}$, where $\mathcal{D}_{i}$ is the Heegaard
diagram of the curves in $C$ with respect to $\Delta_{i}$, such that:

  1. (1)

$\Delta$ = $\Delta_{0}$, $\mathcal{D}$ = $\mathcal{D}_{0}$, $\Delta^{\prime}$
= $\Delta_{n}$, $\mathcal{D}^{\prime}$ = $\mathcal{D}_{n}$.

  2. (2)

For each $i$, $0\leq i<n$, $\Delta_{i+1}$ is obtained from $\Delta_{i}$ by
replacing a disk of $\Delta_{i}$ with a disk whose boundary in $\partial H$ is
a bandsum of the boundaries of the two disks of $\Delta_{i}$ in $\partial H$.

  3. (3)

For each $i$, $0\leq i\leq n$, $\mathcal{D}_{i}$ is connected and has no cut-
vertex.

It then follows, using Lemma 4.3 inductively on $i$, that $\omega$ appears as
a distinguished wave based at $R$ in each Heegaard diagram $\mathcal{D}_{i}$
with $0\leq i\leq n$. In particular, $\omega$ appears in both $\mathcal{D}$ =
$\mathcal{D}_{0}$ and $\mathcal{D}^{\prime}$ = $\mathcal{D}_{n}$ as a
distinguished wave based at $R$. ∎

######  Lemma 4.2.

_Sequences of connected Heegaard diagrams without cut-vertices._

Suppose $\mathcal{C}$ is a finite set of pairwise disjoint simple closed
curves on the boundary of a genus two handlebody $H$, $\Delta$ and
$\Delta^{\prime}$ are two complete sets of meridian disks for $H$, and
$\mathcal{D}$, respectively $\mathcal{D}^{\prime}$, is the Heegaard diagram of
the curves in $\mathcal{C}$ with respect to $\Delta$, respectively
$\Delta^{\prime}$. If both $\mathcal{D}$ and $\mathcal{D}^{\prime}$ are
connected and have no cut-vertex, then there exists a finite sequence
$\\{\Delta_{i}\\}$, $0\leq i\leq n$, of complete sets of meridian disks for
$H$, and an associated sequence of Heegaard diagrams $\\{\mathcal{D}_{i}\\}$,
where $\mathcal{D}_{i}$ is the Heegaard digram of the curves in $C$ with
respect to $\Delta_{i}$, such that:

  1. (1)

$\Delta$ = $\Delta_{0}$, $\mathcal{D}$ = $\mathcal{D}_{0}$, $\Delta^{\prime}$
= $\Delta_{n}$, $\mathcal{D}^{\prime}$ = $\mathcal{D}_{n}$.

  2. (2)

For each $i$, $0\leq i<n$, $\Delta_{i+1}$ is obtained from $\Delta_{i}$ by
replacing a disk of $\Delta_{i}$ with a disk whose boundary in $\partial H$ is
a bandsum of the boundaries of the two disks of $\Delta_{i}$ in $\partial H$.

  3. (3)

For each $i$, $0\leq i\leq n$, $\mathcal{D}_{i}$ is connected and has no cut-
vertex.

###### Proof.

Given $\mathcal{C}$ on $\partial H$, let $min(\mathcal{C})$ consist of the
complete sets of meridian disks of $H$ which intersect the curves in
$\mathcal{C}$ minimally.

Then the goal is to show that a sequence $\\{\Delta_{i}\\}$, of complete sets
of meridian disks for $H$ satisfying the lemma can be constructed by
concatenating up to three subsequences of complete sets of meridian disks for
$H$, depending on whether $\Delta\notin min(\mathcal{C})$, $\Delta\in
min(\mathcal{C})$, $\Delta^{\prime}\notin min(\mathcal{C})$, or
$\Delta^{\prime}\in min(\mathcal{C})$.

Perhaps a detailed example will suffice.

Suppose, for instance, that neither $\Delta$ or $\Delta^{\prime}$ belongs to
$min(\mathcal{C})$. Then, since $\Delta\notin min(\mathcal{C})$, it follows
from Lemma 3.3 that there exists a finite sequence $S_{1}=\\{\Delta_{h}\\}$,
$0\leq h\leq n_{1}$, of complete sets of meridian disks for $H$ such that:
$\Delta_{0}=\Delta$; $\Delta_{n_{1}}\in min(\mathcal{C})$; for each $h$ with
$0\leq h<n_{1}$,
$|\partial\Delta_{h}\cap\mathcal{C}|>|\partial\Delta_{h+1}\cap\mathcal{C}|$;
and, for each $h$ with $0\leq h<n_{1}$, $\Delta_{h+1}$ is obtained from
$\Delta_{h}$ by replacing one of the disks of $\Delta_{h}$ with a disk whose
boundary is a bandsum in $\partial H$ of the boundaries of the two disks in
$\Delta_{h}$.

Furthermore, since, for $0\leq h<n_{1}$,
$|\partial\Delta_{h}\cap\mathcal{C}|>|\partial\Delta_{h+1}\cap\mathcal{C}|$,
we have $|\mathcal{D}_{h}|>|\mathcal{D}_{h+1}|$ in the sequence
$\\{\mathcal{D}_{h}\\}$ of associated Heegaard diagrams, where
$\mathcal{D}_{h}$ is the Heegaard diagram of the curves in $\mathcal{C}$ with
respect to $\Delta_{h}$. Then Lemma 3.5 implies that each $\mathcal{D}_{h+1}$,
$0\leq h<n_{1}$, is obtained from $\mathcal{D}_{h}$ by a bandsum along an edge
of $\mathcal{D}_{h}$. And then Lemma 3.8 or Lemma 3.9 applies and implies each
$\mathcal{D}_{h+1}$, with $0\leq h<n_{1}$, is connected and has no cut-vertex.

Next, since $\Delta^{\prime}\notin min(\mathcal{C})$, the argument used above
for the case in which $\Delta\notin min(\mathcal{C})$, applies again, and it
implies there exists a finite sequence $S_{2}=\\{\Delta_{j}\\}$, $0\leq j\leq
n_{2}$, of complete sets of meridian disks for $H$ such that:
$\Delta_{0}=\Delta^{\prime}$; $\Delta_{n_{2}}\in min(\mathcal{C})$; for each
$j$ with $0\leq j<n_{2}$,
$|\partial\Delta_{j}\cap\mathcal{C}|>|\partial\Delta_{j+1}\cap\mathcal{C}|$;
and, for each $j$ with $0\leq j<n_{2}$, $\Delta_{j+1}$ is obtained from
$\Delta_{j}$ by replacing one of the disks of $\Delta_{j}$ with a disk whose
boundary is a bandsum in $\partial H$ of the boundaries of the two disks in
$\Delta_{j}$.

And again, since, for $0\leq j<n_{2}$,
$|\partial\Delta_{j}\cap\mathcal{C}|>|\partial\Delta_{j+1}\cap\mathcal{C}|$,
we have $|\mathcal{D}_{j}|>|\mathcal{D}_{j+1}|$ in the sequence
$\\{\mathcal{D}_{j}\\}$ of associated Heegaard diagrams, where
$\mathcal{D}_{j}$ is the Heegaard diagram of the curves in $\mathcal{C}$ with
respect to $\Delta_{j}$. Then Lemma 3.5 implies that each $\mathcal{D}_{j+1}$,
$0\leq j<n_{2}$, is obtained from $\mathcal{D}_{j}$ by a bandsum along an edge
of $\mathcal{D}_{j}$. And then Lemma 3.8 or Lemma 3.9 applies and implies each
$\mathcal{D}_{j+1}$, with $0\leq j<n_{2}$, is connected and has no cut-vertex.

Now while $\Delta_{n_{1}}\in min(\mathcal{C})$ and $\Delta_{n_{2}}\in
min(\mathcal{C})$, it may be the case that $\Delta_{n_{1}}$ and
$\Delta_{n_{2}}$ are distinct. In this case, it follows from Lemma 3.4 that
there exists a finite sequence $S_{3}=\\{\Delta_{k}\\}$, $0\leq k\leq n_{3}$,
of complete sets of meridian disks for $H$ such that:
$\Delta_{0}=\Delta_{n_{1}}$; $\Delta_{n_{3}}=\Delta_{n_{2}}$; for each $k$
with $0\leq k<n_{3}$,
$|\partial\Delta_{k}\cap\mathcal{C}|=|\partial\Delta_{k+1}\cap\mathcal{C}|$;
and, for each $k$ with $0\leq k<n_{3}$, $\Delta_{k+1}$ is obtained from
$\Delta_{k}$ by replacing one of the disks of $\Delta_{k}$ with a disk whose
boundary is a bandsum in $\partial H$ of the boundaries of the two disks in
$\Delta_{k}$.

And again, since, for $0\leq k<n_{3}$,
$|\partial\Delta_{k}\cap\mathcal{C}|=|\partial\Delta_{k+1}\cap\mathcal{C}|$,
we have $|\mathcal{D}_{k}|=|\mathcal{D}_{k+1}|$ in the sequence
$\\{\mathcal{D}_{k}\\}$ of associated Heegaard diagrams, where
$\mathcal{D}_{k}$ is the Heegaard diagram of the curves in $\mathcal{C}$ with
respect to $\Delta_{k}$. Then Lemma 3.5 once more implies that each
$\mathcal{D}_{k+1}$, $0\leq k<n_{3}$, is obtained from $\mathcal{D}_{k}$ by a
bandsum along an edge of $\mathcal{D}_{k}$. After which Lemma 3.8 or Lemma 3.9
applies and implies each $\mathcal{D}_{k+1}$, with $0\leq k<n_{3}$, is
connected and has no cut-vertex.

Finally, a sequence of complete sets of meridian disks $\\{\Delta_{i}\\}$ for
$H$ and an associated sequence of connected Heegaard diagrams
$\\{\mathcal{D}_{i}\\}$, without cut-vertices, of the curves in $\mathcal{C}$,
where $\mathcal{D}_{i}$ is the Heegaard diagram of the curves in $\mathcal{C}$
with respect to $\Delta_{i}$, satisfying the statement of the lemma, can be
assembled from $S_{1}$ followed by $S_{3}$ and $-S_{2}$, where $-S_{2}$ is the
sequence of complete sets of meridian disks for $H$ obtained by visiting the
complete sets of meridian disks for $H$ in $S_{2}$ in reverse order. ∎

######  Lemma 4.3.

_Distinguished wave persistence under a bandsum._

Suppose $R$ is a nonseparating simple closed curve on the boundary of a genus
two handlebody $H$, $\\{D_{A},D_{B}\\}$ is a complete set of meridian disks
for $H$, the Heegaard diagram $\mathcal{D}$ of $R$ with respect to
$\\{D_{A},D_{B}\\}$ is connected and has no cut-vertex, and $\omega$ is the
distinguished wave based at $R$ in $\mathcal{D}$. In addition, suppose
$\\{D_{A},D_{C}\\}$ is another complete set of meridian disks of $H$ in which
the boundary $C$ of $D_{C}$ is a bandsum of $\partial D_{A}$ with $\partial
D_{B}$ in $\partial H$, $\mathcal{D}^{\prime}$ is the Heegaard diagram of $R$
with respect to $\\{D_{A},D_{C}\\}$, and either the complexity of
$\mathcal{D}^{\prime}$ is not greater than the complexity of $\mathcal{D}$, or
$\mathcal{D}^{\prime}$ is connected and has no cut-vertex. Then $\omega$ also
appears in $\mathcal{D}^{\prime}$ as a distinguished wave based at $R$.

###### Proof.

It follows from Lemmas 3.5 and 3.6 that the curve $C$ is a bandsum of
$\partial D_{A}$ with $\partial D_{B}$ along an edge of $\mathcal{D}$. Then
there are two cases depending on whether $\mathcal{D}$ is positive or
nonpositive.

First, suppose $\mathcal{D}$ is positive. Then $\mathcal{D}$ and $C$ have the
form shown in Figure 5, and the distinguished wave $\omega$ based at $R$
appears as the horizontal wave $\omega_{h}$ based at $R$.

Since $c>0$ and $d>0$, $C$ has both positive and negative signed intersections
with $R$, and so $\mathcal{D}^{\prime}$ is nonpositive. In addition, since
$\omega_{h}$ is disjoint from $\partial D_{A}$ and $C$ in Figure 5,
$\omega_{h}$ appears in $\mathcal{D}^{\prime}$ as a wave based at $R$. But
waves based at $R$ in nonpositive connected Heegaard diagrams without cut-
vertices are unique. So $\omega_{h}$ appears as the distinguished wave based
at $R$ in $\mathcal{D}^{\prime}$.

Next, suppose $\mathcal{D}$ is nonpositive. In this case, there are two
possibilities depending on whether $\mathcal{D}^{\prime}$ is positive or
nonpositive.

First, suppose $\mathcal{D}^{\prime}$ is nonpositive. Since $\omega$ in
$\mathcal{D}$ is properly isotopic to a subarc of $\partial D_{A}$ or
$\partial D_{B}$ in $\mathcal{D}$, and the curve $C$ is disjoint from
$\partial D_{A}$ and $\partial D_{B}$ in $\mathcal{D}$, $\omega$ appears as a
wave based at $R$ in $\mathcal{D}^{\prime}$. However, since
$\mathcal{D}^{\prime}$ is nonpositive, the distinguished wave based at $R$ is
the only wave based at $R$. So $\omega$ is the distinguished wave based at
$R$.

Second, suppose $\mathcal{D}^{\prime}$ is positive. As in the case in which
$\mathcal{D}^{\prime}$ is nonpositive, $\omega$ appears as a wave based at $R$
in $\mathcal{D}^{\prime}$. But now there is the possibility that $\omega$
appears as a vertical wave $\omega_{v}$ based at $R$ in
$\mathcal{D}^{\prime}$, rather than as a horizontal wave $\omega_{h}$ based at
$R$ in $\mathcal{D}^{\prime}$. So the possibility that $\omega$ appears as a
vertical wave $\omega_{v}$ in $\mathcal{D}^{\prime}$ needs to be ruled out.

To do this, consider Figure 8. In passing from $\mathcal{D}$ to
$\mathcal{D}^{\prime}$, cutting disk $D_{B}$ was discarded, and $D_{B}$ was
replaced with $D_{C}$. But the curve $B=\partial D_{B}$ is still present in
$\mathcal{D}^{\prime}$, and because $\mathcal{D}$ is connected, and has no
cut-vertex, $B$ must appear in $\mathcal{D}^{\prime}$ as a bandsum of
$\partial D_{A}$ with $\partial D_{C}$ along an edge of
$\mathcal{D}^{\prime}$. So $B$ must appear in Figure 8 as shown, in which case
$B$ has an essential intersection with $\omega_{v}$ in $\mathcal{D}^{\prime}$.
Conclude that, since $\omega$ is disjoint from $B=\partial D_{B}$ in
$\mathcal{D}$, $\omega$ appears as the horizontal wave $\omega_{h}$ based at
$R$ in $\mathcal{D}^{\prime}$. ∎

Figure 8. Bandsums along edges of positive diagrams yield connected diagrams
without cut-vertices in which the distinguished wave $\boldsymbol{\omega_{h}}$
persists. Here $a,c,d>0$.

######  Definition 4.4.

Distinguished-wave-determined slope representatives
and distinguished-wave-determined slopes.

Suppose $R$ is a nonseparating simple closed curve on the boundary of a genus
two handlebody $H$, $H[R]$ has incompressible boundary, $\Delta$ is a complete
set of meridian disks for $H$ such that the Heegaard diagram $\mathcal{D}$ of
$R$ with respect to $\Delta$ is connected, has no cut-vertex, and $\omega$ is
the distinguished wave based at $R$ in $\mathcal{D}$. Then the wave-determined
slope representatives of Definition 2.3 are _distinguished-wave-determined
slope representatives_ , and wave-determined slopes of Definition 2.4 as
_distinguished-wave-determined slopes_. (Usually shortened to _distinguished-
slope representatives_ and _distinguished-slopes_ respectively.)

######  Lemma 4.5.

_Distinguished-wave-determined slope representatives never bound disks._

Suppose $R$ is a nonseparating simple closed curve on the boundary of a genus
two handlebody $H$, $\Delta$ is a complete set of meridian disks for $H$ such
that the Heegaard diagram $\mathcal{D}$ of $R$ with respect to $\Delta$ is
connected, has no cut-vertex, $\omega$ is the distinguished wave based at $R$
in $\mathcal{D}$, and $m_{1}$ and $m_{2}$ are the two distinguished slope
representatives obtained by surgery on $R$ along $\omega$. Then neither
$m_{1}$ or $m_{2}$ bound disks in $H$.

###### Proof.

If $m_{1}$ or $m_{2}$ bounds a disk in $H$, $\mathcal{D}$ is either not
connected, or has a cut-vertex, contrary to hypothesis. ∎

##  5\. Only distinguished-wave-determined slopes are meridians of tunnel-
number-one knots in $S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times
S^{2})\;\\#\;L(p,q)$.

######  Lemma 5.1.

_Waves provide meridians._

Suppose H is a genus two handlebody, R is a nonseparating simple closed curve
on $\partial H$ such that the manifold $H[R]$ has incompressible boundary, and
$H[R]$ is the exterior of a knot in $S^{3}$, $S^{1}\times S^{2}$, or
$(S^{1}\times S^{2})\;\\#\;L(p,q)$. Then a simple closed curve $m$ in
$\partial H$ representing the meridian of $H[R]$ can be obtained by surgery on
$R$ along a wave based at $R$.

###### Proof.

Since $H[R]$ has incompressible boundary, there exists a complete set of
meridian disks $\\{D_{A},D_{B}\\}$ for $H$ such that the Heegaard diagram
$\mathcal{D}$ of $R$ with respect to $\\{D_{A},D_{B}\\}$ is connected and has
no cut-vertex. And, since $H[R]$ is the exterior of a knot in $S^{3}$,
$S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$, there are simple
closed curves in $\partial H$, disjoint from $R$, which represent the meridian
of $H[R]$. Among all such representatives of the meridian of $H[R]$, suppose
$m$ is chosen so that the complexity of the Heegaard diagram
$\mathcal{D}^{\prime}$ of $\\{m,R\\}$ with respect to $\\{D_{A},D_{B}\\}$ is
minimal. Then, since $\mathcal{D}$ is connected and has no cut-vertex,
$\mathcal{D}^{\prime}$ is also connected, has nlabelo cut-vertex, and
certainly has complexity greater than two.

It now follows from the main results of [HOT80] and [NO85] that
$\mathcal{D}^{\prime}$ must have a wave $\omega$ based at one of $\partial
D_{A}$, $\partial D_{B}$, $m$, or $R$. However, since $\mathcal{D}^{\prime}$
is connected, and has no cut-vertex, $\omega$ can not be based at $\partial
D_{A}$ or $\partial D_{B}$. Furthermore, the minimality condition used to
choose $m$ implies $\omega$ is not based at $m$. So $\omega$ must be based at
$R$. Then, since the boundary components of a regular neighborhood $N$ of
$R\cup\omega$ in $\partial H$ form a pant decomposition of $\partial H$, $m$
must be isotopic to one of the two boundary components of $N$ not isotopic to
$R$ in $\partial H$. In other words, $m$ can be obtained by surgery on $R$
along a wave $\omega$ based at $R$. ∎

######  Theorem 5.2.

_Only distinguished-wave-determined slopes are meridians of tunnel-number-one
knots in $\boldsymbol{S^{3}}$, $\boldsymbol{S^{1}\times S^{2}}$, or
$\boldsymbol{(S^{1}\times S^{2})\;\\#\;L(p,q)}$._

Suppose $R$ is a nonseparating simple closed curve on the boundary of a genus
two handlebody $H$ such that $R$ has a connected Heegaard diagram
$\mathcal{D}$ without cut-vertices with respect to a complete set of meridian
disks $\\{D_{A},D_{B}\\}$ of $H$. Then only Dehn filling of $H[R]$ along the
distinguished-wave-determined slope can yield $S^{3}$, $S^{1}\times S^{2}$, or
$(S^{1}\times S^{2})\;\\#\;L(p,q)$.

###### Proof.

Lemma 5.1 shows that, if $H[R]$ embeds in $S^{3}$, $S^{1}\times S^{2}$, or
$(S^{1}\times S^{2})\;\\#\;L(p,q)$, a representative of the meridian of $H[R]$
can be obtained by surgery on $R$ along a wave $\omega$ based at $R$ in
$\mathcal{D}$. It remains to show that in this case $\omega$ is actually a
distinguished wave based at $R$.

There are two cases to consider, depending upon whether the Heegaard diagram
$\mathcal{D}$ of $R$ with respect to $\\{D_{A},D_{B}\\}$ is positive or
nonpositive. If $\mathcal{D}$ is nonpositive, Lemma 3.10 shows $\omega$ is the
unique wave based at $R$ in $\mathcal{D}$, and so, in this case, $\omega$ is
the distinguished wave based at $R$ by default.

The case in which $\mathcal{D}$ is positive requires some work, since in this
case there are always two distinct waves based at $R$ in $\mathcal{D}$, only
one of which is a distinguished wave. In particular, when $\mathcal{D}$ is
connected, positive and has no cut-vertex, the graph of $\mathcal{D}$ has the
form of Figure 7a in 3.12, and $\omega$ is either the wave $\omega_{h}$ or the
wave $\omega_{v}$ in Figure 7b in 3.12.

Lemma 5.3 below rules out the possibility that $\omega$ is a vertical wave
$\omega_{v}$, as in Figure 7b in 3.12. ∎

######  Lemma 5.3.

_Only horizontal waves yield meridians of positive diagrams of
$\boldsymbol{S^{3}}$, $\boldsymbol{S^{1}\times S^{2}}$, or
$\boldsymbol{(S^{1}\times S^{2})\;\\#\;L(p,q)}$._

Suppose $R$ is a nonseparating simple closed curve on the boundary of a genus
two handlebody $H$, $R$ has a connected, positive Heegaard diagram without
cut-vertices with respect to a complete set of meridian disks
$\\{D_{A},D_{B}\\}$ of $H$, and $H[R]$ embeds in $S^{3}$, $S^{1}\times S^{2}$,
or $(S^{1}\times S^{2})\;\\#\;L(p,q)$. Then only surgery on $R$ along a
horizontal wave $\omega_{h}$ based at $R$ will yield a representative of the
meridian of $H[R]$.

###### Proof.

The hypotheses imply that $R$ has a genus two Heegaard diagram of the form
shown in Figure 7a with parameters $a,c,d>0$. By Lemma 5.1, if $H[R]$ embeds
in $S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$, a
representative $m$ of the meridian of $H[R]$ can be obtained by surgery on $R$
along a wave $\omega$ in the Heegaard diagram of $R$ with respect to the set
of meridian disks $\\{D_{A},D_{B}\\}$ of $H$. As Figure 7b shows, there are
two such waves, $\omega_{v}$ and $\omega_{h}$ along which $R$ can be surgered
to obtain a candidate for the meridian of $H[R]$. We want to rule out the
possibility that a representative $m$ of the meridian of $H[R]$ can be
obtained by surgery on $R$ along the vertical wave $\omega_{v}$.

To do this, start with the enhanced version of Figure 7b displayed in Figure
9. Figure 9 is a positive, connected Heegaard diagram $\mathcal{D}$, without
cut-vertices, of a nonseparating simple closed curve $R$ with respect to a
complete set of meridian disks $\\{D_{A},D_{B}\\}$ of a genus two handlebody
$H$. In addition, Figure 9 displays a vertical wave $\omega_{v}$ based at $R$,
along with the arcs in which the curves $m_{1}$ and $m_{2}$, obtained by
surgery on $R$ along $\omega_{v}$, intersect the two major non-rectangular
faces, say $F^{+}$ and $F^{-}$, of $\mathcal{D}$. Also, a curve $C$, which is
a bandsum of $\partial D_{A}$ and $\partial D_{B}$ along an edge of
$\mathcal{D}$, has been added to Figure 9.

Next, supposing $m$ has been chosen to be one of $m_{1}$, $m_{2}$ in Figure 9,
let $D_{C}$ be the meridian disk of $H$, such that $\partial D_{C}$ = $C$ in
Figure 9, let $\\{D_{A},D_{C}\\}$ be the complete set of meridian disks of
$H$, obtained by replacing the meridian disk $D_{B}$ of $H$ with the meridian
disk $D_{C}$ of $H$, and let $\mathcal{D}^{\prime}$ be the Heegaard diagram of
$m$ and $R$ with respect to $\\{D_{A},D_{C}\\}$.

Observe that, since the curve $C$ in Figure 9 is a bandsum of $\partial D_{A}$
with $\partial D_{B}$ along an edge of $\mathcal{D}$, the subdiagram of
$\mathcal{D}^{\prime}$ consisting of the Heegaard diagram of $R$ with respect
to $\\{D_{A},D_{C}\\}$ is connected, and has no cut-vertex. It follows
$\mathcal{D}^{\prime}$ is also connected, and has no cut-vertex.

Next, recall that the main results of [HOT80] and [NO85] show that if $m$
represents the meridian of $H[R]$, then any connected, genus two Heegaard
diagram, without cut-vertices, of $\\{R,m\\}$ must have a wave based at $R$ or
$m$. We show this is not the case by observing that the Heegaard diagram
$\mathcal{D}^{\prime}$ of $m$ and $R$ with respect to $\\{D_{A},D_{C}\\}$ does
not have a wave based at $R$ or $m$.

To see that the Heegaard diagram $\mathcal{D}^{\prime}$ of $m$ and $R$ with
respect to $\\{D_{A},D_{C}\\}$ does not have a wave based at $R$ or $m$,
return to Figure 9, and observe that $C\cap F^{+}$ is a single arc $e^{+}$,
and $C\cap F^{-}$ is a single arc $e^{-}$, while each of $m_{1}\cap e^{+}$,
$m_{1}\cap e^{-}$, $m_{2}\cap e^{+}$, and $m_{2}\cap e^{-}$ is a single point.

Then consider Figure 10 which displays a twice-punctured torus $F$ obtained by
cutting the Heegaard surface $\partial H$ of Figure 9 open along the curve
$R$. The two arcs $e^{+}$ and $e^{-}$ in which the curve $C$ of Figure 9
intersects the two major non-rectangular faces of the Heegaard diagram
$\mathcal{D}$ are also shown in this figure, along with the two curves $m_{1}$
and $m_{2}$ obtained by surgery on $R$ along the vertical wave $\omega_{v}$ in
Figure 9.

Next, observe that any properly embedded essential arc, say $w$, in $F$, which
is a candidate for a wave based at $R$ in $\mathcal{D}^{\prime}$, must be
properly isotopic in $F$ to one of $e^{+}$, $e^{-}$. However, then $w$ has
essential intersections with both $m_{1}$ and $m_{2}$ in $F$, so $w$ has
essential intersections with $m$ in $\mathcal{D}^{\prime}$, and $w$ is not a
wave in $\mathcal{D}^{\prime}$. Conclude there is no wave based at $R$ in
$\mathcal{D}^{\prime}$.

Finally, observe that cutting $F$ open along $e^{+}$ and $e^{-}$ yields two
annuli, say $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$, and that $m_{1}$ and
$m_{2}$ each intersect each of these annuli in a single essential arc. Any
wave based at $m$ in $\mathcal{D}^{\prime}$ must be based at one of $m_{1}$,
$m_{2}$ and lie in one of $\mathcal{A}_{1}$, $\mathcal{A}_{2}$. Clearly no
such wave exists. Conclude there is no wave based at $m$ in
$\mathcal{D}^{\prime}$. This finishes the proof of Theorem 5.3. ∎

Below are two immediate corollaries. Among other things, the first corollary
implies the tunnel-number-one knot case of Gabai’s result [G87] that surgery
on a knot in $S^{3}$ never yields $S^{1}\times S^{2}$, while the second
corollary implies the tunnel-number-one knot case of Gordon and Luecke’s
result [GL89] that knots in $S^{3}$ are determined by their complements.

######  Corollary 5.4.

$H[R]$ embeds in at most one of $S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times
S^{2})\;\\#\;L(p,q)$.

######  Corollary 5.5.

Tunnel-number-one knots in $S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times
S^{2})\;\\#\;L(p,q)$ are determined by their complements.

Figure 9.  Figure 10.

##  6\. Distinguished waves in genus two Heegaard diagrams of closed
manifolds.

######  Theorem 6.1.

_Distinguished waves in genus two Heegaard diagrams of closed manifolds are
well-defined._

Suppose $R_{1}$ and $R_{2}$ are disjoint, nonparallel, nonseparating simple
closed curves in the boundary of a genus two handlebody $H$, $\Delta$ and
$\Delta^{\prime}$ are two complete sets of meridian disks for $H$, and both
the Heegaard diagram $\mathcal{D}$ of $R_{1}$ and $R_{2}$ with respect to
$\Delta$, and the Heegaard diagram $\mathcal{D}^{\prime}$ of $R_{1}$ and
$R_{2}$ with respect to $\Delta^{\prime}$ are connected and have no cut-
vertices. In addition, suppose the subdiagram $\mathcal{D}^{*}$ of
$\mathcal{D}$, consisting of the edges of $R_{1}$ in $\mathcal{D}$, is
connected, has no cut-vertex, and $\omega$ is a distinguished wave based at
$R_{1}$ in $\mathcal{D}$. Then the subdiagram $\mathcal{D}^{\prime*}$ of
$\mathcal{D}^{\prime}$, consisting of the edges of $R_{1}$ in
$\mathcal{D}^{\prime}$, is also connected, has no cut-vertex, and $\omega$ is
a distinguished wave based at $R_{1}$ in $\mathcal{D}^{\prime}$.

###### Proof.

Since $\mathcal{D}$ and $\mathcal{D}^{\prime}$ are both connected Heegaard
diagrams without cut-vertices, Lemma 4.2 applies, and it shows there exists a
finite sequence $\\{\Delta_{i}\\}$, $0\leq i\leq n$, of complete sets of
meridian disks for $H$, and an associated sequence of Heegaard diagrams
$\\{\mathcal{D}_{i}\\}$, where $\mathcal{D}_{i}$ is the Heegaard digram of
$R_{1}$ and $R_{2}$ with respect to $\Delta_{i}$, such that:

  1. (1)

$\Delta_{0}$ = $\Delta$, $\mathcal{D}_{0}$ = $\mathcal{D}$, $\Delta_{n}$ =
$\Delta^{\prime}$, $\mathcal{D}_{n}$ = $\mathcal{D}^{\prime}$;

  2. (2)

For each $i$, $0\leq i<n$, $\Delta_{i+1}$ is obtained from $\Delta_{i}$ by
replacing a disk of $\Delta_{i}$ with a disk whose boundary in $\partial H$ is
a bandsum of the boundaries of the two disks of $\Delta_{i}$ in $\partial H$;

  3. (3)

For each $i$, $0\leq i\leq n$, $\mathcal{D}_{i}$ is connected and has no cut-
vertex.

Now Lemma 6.2 below shows that more is true. Namely, the subdiagram of each
$\mathcal{D}_{i}$, $0\leq i\leq n$, consisting of the edges of
$\mathcal{D}_{i}$ belonging to $R_{1}$, is connected and has no cut-vertex.
This implies $\omega$ appears as a distinguished wave based at $R_{1}$ in each
$\mathcal{D}_{i}$ for $0\leq i\leq n$. In particular, $\omega$ appears as a
distinguished wave based at $R_{1}$ in $\mathcal{D}$. ∎

######  Lemma 6.2.

_Distinguished waves persist under bandsums that preserve connectivity and no
cut-vertices when two relators are present._

Suppose $R_{1}$ and $R_{2}$ are disjoint nonseparating simple closed curves on
the boundary of a genus two handlebody $H$, $\Delta$ is a complete set of
meridian disks for $H$, the Heegaard diagram $\mathcal{D}$ of $R_{1}\cup
R_{2}$ with respect to $\Delta$ is connected and has no cut-vertex, and the
subdiagram $\mathcal{D}^{*}$ of $\mathcal{D}$ consisting of the edges of
$R_{1}$ in $\mathcal{D}$ is also connected and has no cut-vertex.

In addition, suppose $\omega$ is a distinguished wave based at $R_{1}$ in
$\mathcal{D}$, and $\Delta^{\prime}$ is a complete set of meridian disks for
$H$, obtained by replacing one member of $\Delta$ with a disk whose boundary
is a bandsum $\beta$ in $\partial H$ of the boundaries of the disks of
$\Delta$, such that the Heegaard diagram $\mathcal{D}^{\prime}$ of $R_{1}\cup
R_{2}$ with respect to $\Delta^{\prime}$ is connected and has no cut-vertex.

Then the subdiagram $\mathcal{D}^{\prime\prime}$ of $\mathcal{D}^{\prime}$
consisting of the edges of $R_{1}$ in $\mathcal{D}^{\prime}$ is connected, has
no cut-vertex, and $\omega$ is a distinguished wave based at $R_{1}$ in
$\mathcal{D}^{\prime}$.

###### Proof.

Lemma 3.6 implies that, because $\mathcal{D}^{\prime}$ is connected and has no
cut-vertex, the bandsum $\beta$ must be along a band of parallel edges of
$\mathcal{D}$. Now there are two possibilities depending on whether
$\mathcal{D}^{*}$ is nonpositive or positive.

If $\mathcal{D}^{*}$ is nonpositive, $\beta$ is also a bandsum along an edge
of $\mathcal{D}^{*}$, and it follows from Lemma 3.8 that
$\mathcal{D}^{\prime\prime}$ is connected and has no cut-vertex. But then
$\omega$ is a distinguished wave based at $R_{1}$ in $\mathcal{D}^{\prime}$,
per Lemma 4.3.

So, suppose $\mathcal{D}^{*}$ is positive. In this case $\omega$ must be
horizontal. Then, because $R_{2}$ is disjoint from $\omega$, $\mathcal{D}$
must also be positive. This implies $\beta$ is again a bandsum along an edge
of $\mathcal{D}^{*}$. Then Lemma 3.9 shows $\mathcal{D}^{\prime\prime}$ is
connected and has no cut-vertex. So again $\omega$ is a distinguished wave
based at $R_{1}$ in $\mathcal{D}^{\prime}$, per Lemma 4.3. ∎

######  Lemma 6.3.

_At most one distinguished-wave in a two-relator diagram._

Suppose $R_{1}$ and $R_{2}$ are disjoint, nonparallel, nonseparating simple
closed curves on the boundary of a genus two handlebody $H$, $\Delta$ is a
complete set of meridian disks for $H$, $\mathcal{D}$ is the Heegaard diagram
of $R_{1}\cup R_{2}$ on $\partial H$ with respect to $\Delta$, and the
subdiagrams of $\mathcal{D}$, consisting of the edges of $R_{1}$ in
$\mathcal{D}$, respectively the edges of $R_{2}$ in $\mathcal{D}$, are each
connected and have no cut-vertex. Then either there is no distinguished wave
based at $R_{1}$ in $\mathcal{D}$, or there is no distinguished wave based at
$R_{2}$ in $\mathcal{D}$.

###### Proof.

Suppose $\omega_{1}$ is a distinguished wave based at $R_{1}$ in
$\mathcal{D}$, and let $m_{1}$ and $m_{2}$ be the two distinguished curves
obtained by surgery on $R_{1}$ along $\omega_{1}$. Then the complexities
$|m_{1}|$, $|m_{2}|$, and $|R_{1}|$ of $m_{1}$, $m_{2}$ and $R_{1}$ in
$\mathcal{D}$ satisfy $|m_{1}|<|R_{1}|$ and $|m_{2}|<|R_{1}|$.

Since $R_{2}$ and $\omega_{1}$ are disjoint, $R_{2}$ is isotopic in $\partial
H$ to one of $m_{1}$, $m_{2}$. So $|R_{2}|<|R_{1}|$ in $\mathcal{D}$. This
implies there is no distinguished wave $\omega_{2}$ based at $R_{2}$ in
$\mathcal{D}$. ∎

##  7\. Distinguished waves in genus two Heegaard diagrams of $S^{3}$,
$S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$.

######  Theorem 7.1.

_A distinguished wave appears in each connected genus two Heegaard diagram,
without cut-vertices, of $\boldsymbol{(S^{1}\times S^{2})\;\\#\;L(p,q)}$,
$\boldsymbol{S^{3}}$, or $\boldsymbol{S^{1}\times S^{2}}$._

Suppose $R_{1}$ and $R_{2}$ are disjoint, nonseparating simple closed curves
on the boundary of a genus two handlebody $H$, $\\{D_{A},D_{B}\\}$ is a
complete set of meridian disks for $H$, and the Heegaard diagram $\mathcal{D}$
of $\\{R_{1},R_{2}\\}$ with respect to $\\{D_{A},D_{B}\\}$ is a connected
Heegaard diagram, without cut-vertices, of $S^{3}$, $S^{1}\times S^{2}$, or
$(S^{1}\times S^{2})\;\\#\;L(p,q)$. Then there is a distinguished wave
$\omega$ in $\mathcal{D}$ based at one of $R_{1},R_{2}$, and if $\omega$ is
based at $R_{1}$, respectively $R_{2}$, the subdiagram of $\mathcal{D}$
consisting of the edges of $\mathcal{D}$ belonging to $R_{1}$, respectively
$R_{2}$, is connected and has no cut-vertex.

###### Proof.

Lemma 7.2 shows that one of $R_{1}$, $R_{2}$, say $R_{1}$, is not primitive or
a proper-power in $H$. Then it follows from Lemma 3.2 that there exists a
complete set of meridian disks $\\{D_{A}^{\prime},D_{B}^{\prime}\\}$ of $H$
and a Heegaard diagram $\mathcal{D}^{\prime}$ of $\\{R_{1},R_{2}\\}$ with
respect to $\\{D_{A}^{\prime},D_{B}^{\prime}\\}$ such that the subdiagram of
$R_{1}$ with respect to $\\{D_{A}^{\prime},D_{B}^{\prime}\\}$ is connected and
has no cut-vertex. Next, ignoring $R_{2}$ for the moment, Theorem 5.2 implies
there exists a distinguished wave $\omega^{\prime}$ based at $R_{1}$ in
$\mathcal{D}^{\prime}$.

However, $R_{2}$ is present in $\mathcal{D}^{\prime}$, and there are two
possibilities: either $R_{2}$ is disjoint from $\omega^{\prime}$, or $R_{2}$
and $\omega^{\prime}$ have essential intersections.

Suppose first that $R_{2}$ is disjoint from $\omega^{\prime}$. Then, because
the subdiagram of $R_{1}$ with respect to
$\\{D_{A}^{\prime},D_{B}^{\prime}\\}$ is connected and has no cut-vertex,
$\mathcal{D}^{\prime}$ is also connected and has no cut-vertex. Then Lemma 4.2
applies and it shows there exists a finite sequence $\\{\Delta_{i}\\}$, $0\leq
i\leq n$, of complete sets of meridian disks for $H$, and an associated
sequence of Heegaard diagrams $\\{\mathcal{D}_{i}\\}$, where $\mathcal{D}_{i}$
is the Heegaard digram of $R_{1}$ and $R_{2}$ with respect to $\Delta_{i}$,
such that:

  1. (1)

$\Delta_{0}$ = $\\{D_{A}^{\prime},D_{B}^{\prime}\\}$, $\mathcal{D}_{0}$ =
$\mathcal{D}^{\prime}$, $\Delta_{n}$ = $\\{D_{A},D_{B}\\}$, $\mathcal{D}_{n}$
= $\mathcal{D}$;

  2. (2)

For each $i$, $0\leq i<n$, $\Delta_{i+1}$ is obtained from $\Delta_{i}$ by
replacing a disk of $\Delta_{i}$ with a disk whose boundary in $\partial H$ is
a bandsum of the boundaries of the two disks of $\Delta_{i}$ in $\partial H$;

  3. (3)

For each $i$, $0\leq i\leq n$, $\mathcal{D}_{i}$ is connected and has no cut-
vertex.

Now Lemma 6.2 shows that more is true. Namely, the subdiagram of each
$\mathcal{D}_{i}$, $0\leq i\leq n$, consisting of the edges of
$\mathcal{D}_{i}$ belonging to $R_{1}$, is connected and has no cut-vertex.
This implies $\omega^{\prime}$ appears as a distinguished wave based at
$R_{1}$ in each $\mathcal{D}_{i}$ for $0\leq i\leq n$. In particular,
$\omega^{\prime}$ appears as a distinguished wave based at $R_{1}$ in
$\mathcal{D}$. So, in this case, $\omega^{\prime}$ is the desired
distinguished wave $\omega$ in $\mathcal{D}$.

Next, suppose $R_{2}$ has essential intersections with $\omega^{\prime}$ in
$\mathcal{D}^{\prime}$. Then Lemma 8.1 applies and shows that the subdiagram
of $\mathcal{D}^{\prime}$ consisting of the edges of $R_{2}$ in
$\mathcal{D}^{\prime}$ is connected and has no cut-vertex. So there exists a
subarc $\omega^{\prime\prime}$ of $\omega^{\prime}$, such that
$\omega^{\prime\prime}$ is a wave based at $R_{2}$ in $\mathcal{D}^{\prime}$,
and $R_{1}$, disjoint from $\omega^{\prime\prime}$ represents the meridian of
$H[R_{2}]$. It follows from Theorem 5.2, that $\omega^{\prime\prime}$ is a
distinguished wave based at $R_{2}$ in $\mathcal{D}^{\prime}$.

Finally, since $R_{1}$ and $\omega^{\prime\prime}$ are disjoint in
$\mathcal{D}^{\prime}$, the argument used above, when $\omega^{\prime}$ is
disjoint from $R_{2}$, works with $R_{1}$ and $\omega^{\prime}$ swapped with
$R_{2}$ and $\omega^{\prime\prime}$, to show that $\omega^{\prime\prime}$,
based at $R_{2}$, is the desired distinguished wave $\omega$ in $\mathcal{D}$.
∎

######  Lemma 7.2.

_Connected genus two Heegaard diagrams of $\boldsymbol{S^{3}}$,
$\boldsymbol{S^{1}\times S^{2}}$ or $\boldsymbol{(S^{1}\times
S^{2})\;\\#\;L(p,q)}$, without cut-vertices, have a relator which is not
primitive or a proper-power._

Suppose $R_{1}$ and $R_{2}$ are disjoint nonseparating simple closed curves on
the boundary of a genus two handlebody $H$, $\\{D_{A},D_{B}\\}$ is a complete
set of meridian disks for $H$, the Heegaard diagram $\mathcal{D}$ of
$\\{R_{1},R_{2}\\}$ with respect to $\\{D_{A},D_{B}\\}$ is connected, has no
cut-vertex, and is a Heegaard diagram of $S^{3}$, $S^{1}\times S^{2}$, or
$(S^{1}\times S^{2})\;\\#\;L(p,q)$. Then at least one of $R_{1},R_{2}$ is not
primitive or a proper-power in $H$.

###### Proof.

Suppose first, to the contrary, that both $R_{1}$ and $R_{2}$ are primitive in
$H$. Then, since $R_{1}$ is primitive in $H$, there exists a complete set of
meridian disks $\\{D_{A}^{\prime},D_{B}^{\prime}\\}$ of $H$ with the property
that $R_{1}$ intersects $\partial D_{A}^{\prime}$ in a single point of
transverse intersection, and $R_{1}$ is disjoint from $D_{B}^{\prime}$.
Furthermore, among all complete sets of meridian disks of $H$, with
$R_{1}\cap\,\partial D_{A}^{\prime}$ a single point of transverse
intersection, and $R_{1}\cap\,\partial D_{B}^{\prime}$ = $\emptyset$, suppose
$R_{2}\cap\partial D_{A}^{\prime}$ is minimal. This implies that the Heegaard
diagram of $R_{2}$ with respect to $\\{\partial D_{A}^{\prime},\partial
D_{B}^{\prime}\\}$ has no loops. And then, because $R_{2}$ is primitive in
$H$, all of the intersections of $R_{2}$ with $\partial D_{B}$ must have the
same sign.

So suppose $|R_{2}\cap\partial D_{B}|=n\geq 0$. Then, since $\mathcal{D}$ is a
Heegaard diagram of $S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times
S^{2})\;\\#\;L(p,q)$, it must be the case that $n\leq 1.$

However, if $n=0$, $\partial D_{B}$ is disjoint from $R_{1}$ and $R_{2}$,
which implies $\mathcal{D}$ is either not connected, or has a cut-vertex,
contrary to hypothesis. While, if $n=1$, the boundary $\Gamma$ of a regular
neighborhood of $R_{2}\cup\partial D_{B}$ in $\partial H$ is an essential
separating simple closed curve in $\partial H$ bounding a disk in $H$ such
that $\Gamma$ is disjoint from $R_{1}$ and $R_{2}$. This again implies
$\mathcal{D}$ is either not connected, or has a cut-vertex, contrary to
hypothesis.

The remaining possibilities to be ruled out are that one member of
$\\{R_{1},R_{2}\\}$ is primitive in $H$, while the other member of
$\\{R_{1},R_{2}\\}$ is a proper-power in $H$, or that both members of
$\\{R_{1},R_{2}\\}$ are proper-powers in $H$. It is shown in [B09] that in
both of these cases, either there exists an essential separating simple closed
curve $\Gamma$ in $\partial H$, disjoint from $R_{1}$ and $R_{2}$, such that
$\Gamma$ bounds a disk in $H$, or $R_{1}$ and $R_{2}$ are the two boundary
components of a nonseparating annulus $\mathcal{A}$ in $H$.

In the latter case, boundary compression of $\mathcal{A}$ in $H$ yields a
nonseparating disk in $H$, disjoint from $R_{1}$ and $R_{2}$. It then follows,
as above, that each of these possibilities implies $\mathcal{D}$ is either not
connected, or has a cut-vertex, contrary to hypothesis. ∎

##  8\. Some properties of meridian representatives of tunnel-number-one knots
in $S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$.

######  Lemma 8.1.

_If $\boldsymbol{m}$ is a meridian representative whose subdiagram is not
connected or has a cut-vertex in a connected Heegaard diagram, without cut-
vertices, of $\boldsymbol{S^{3}}$, $\boldsymbol{S^{1}\times S^{2}}$, or
$\boldsymbol{(S^{1}\times S^{2})\;\\#\;L(p,q)}$, then $\boldsymbol{m}$ is
distinguished._

Suppose $m$ and $R$ are disjoint nonseparating simple closed curves in the
boundary of a genus two handlebody $H$, $H[R]$ has incompressible boundary,
$H[R]$ embeds in $S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times
S^{2})\;\\#\;L(p,q)$, and $m$ represents the meridian of $H[R]$. In addition,
suppose $\\{D_{A},D_{B}\\}$ is a complete set of meridian disks for $H$ such
that the Heegaard diagram $\mathcal{D}$ of $R$ with respect to
$\\{D_{A},D_{B}\\}$ is connected and has no cut-vertex, $\omega$ is the
distinguished wave based at $R$ in $\mathcal{D}$, $m_{1}$ and $m_{2}$ are the
pair of distinguished meridian representatives of $H[R]$ obtained by surgery
on $R$ along $\omega$, and the Heegaard diagram $\mathcal{D}^{\prime}$ of $m$
with respect to $\\{D_{A},D_{B}\\}$ is not connected, or has a cut-vertex.
Then $m$ is isotopic in $\partial H$ to one of $m_{1},m_{2}$.

###### Proof.

Suppose $m$, $R$, $\partial D_{A}$, and $\partial D_{B}$ are all oriented.
Then, focusing first on the Heegaard diagram $\mathcal{D}^{\prime}$ of $m$
with respect to $\\{D_{A},D_{B}\\}$, note that $m$ cannot have any loops in
$\mathcal{D}^{\prime}$, since the existence of loops in $\mathcal{D}^{\prime}$
would force $\mathcal{D}$ to have a cut-vertex, contrary to hypothesis. It
follows, since $\mathcal{D}^{\prime}$ is loopless, but $\mathcal{D}^{\prime}$
is not connected, or $\mathcal{D}^{\prime}$ has a cut-vertex, that the signed
intersections of $m$ with $\partial D_{A}$ all have the same sign, the signed
intersections of $m$ with $\partial D_{B}$ all have the same sign, and there
are either no edges of $m$ connecting vertices $A^{+}$ and $A^{-}$ of
$\mathcal{D}^{\prime}$, or there are no edges of $m$ connecting vertices
$B^{+}$ and $B^{-}$ of $\mathcal{D}^{\prime}$.

Next, note that, since $m$ is disjoint from $R$ and $m$ is a meridian
representative of $H[R]$, $m$ must intersect the distinguished wave $\omega$
based at $R$ algebraically zero times.

Now there are two cases:

  1. (1)

$R$ has both positive and negative signed intersections with one of $\partial
D_{A}$, $\partial D_{B}$.

  2. (2)

The intersections of $R$ with $\partial D_{A}$ all have the same sign, and the
intersections of $R$ with $\partial D_{B}$ all have the same sign.

First, suppose Case 1 holds. Then there are intersections of $R$ with at least
one of $\partial D_{A}$, $\partial D_{B}$ which have different signs and are
adjacent along $\partial D_{A}$ or $\partial D_{B}$. In this case, the
distinguished wave $\omega$ based at $R$ is isotopic to a subarc of $\partial
D_{A}$ or $\partial D_{B}$. But then $|m\cap\,\omega|=0$. So $m$ is disjoint
from $\omega$, which means $m$ is isotopic to one of $m_{1}$, $m_{2}$ in
$\partial H$.

Next, suppose Case 2 holds. In this case, the diagram $\mathcal{D}$ of $R$ is
positive, and the distinguished wave $\omega$ based at $R$ is a horizontal
wave $\omega_{h}$ in $\mathcal{D}$, as in Figure 7b. Now there are at most two
bands of edges of $m$ in $\mathcal{D}^{\prime}$ which intersect $\omega$, and
these bands have a common vertex of $\mathcal{D}^{\prime}$ as endpoint. It
follows that all of the intersections of $m$ with $\omega$ have the same sign.
So again $m$ must be disjoint from $\omega$, and hence isotopic to one of
$m_{1}$, $m_{2}$ in $\partial H$. ∎

######  Corollary 8.2.

_At most two meridian representatives in diagrams of $\boldsymbol{S^{3}}$,
$\boldsymbol{S^{1}\times S^{2}}$, or $\boldsymbol{(S^{1}\times
S^{2})\;\\#\;L(p,q)}$ are primitive or proper-powers._

Suppose $m$ and $R$ are disjoint nonseparating simple closed curves on the
boundary of a genus two handlebody $H$ such that $H[R]$ has incompressible
boundary, $H[R]$ embeds in $S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times
S^{2})\;\\#\;L(p,q)$, $m$ is primitive or a proper-power in $H$, and $m$
represents the meridian of $H[R]$. Then $m$ is isotopic in $\partial H$ to one
of the two distinguished meridian representatives of $H[R]$. In particular,
there are at most two meridian representatives of $H[R]$ that are primitive or
proper-powers in $H$.

###### Proof.

This follows immediately from Lemma 8.1 since, if $m$ is primitive or a
proper-power in $H$, the Heegaard diagram $\mathcal{D}^{\prime}$ of $m$ with
respect to $\\{D_{A},D_{B}\\}$ is either not connected, or has a cut-vertex.
So $m$ satisfies the hypothesis of Lemma 8.1. ∎

######  Lemma 8.3.

_The two distinguished meridian representatives are the two “shortest”
meridian representatives of a tunnel of a tunnel-number-one knot in
$\boldsymbol{S^{3}}$, $\boldsymbol{S^{1}\times S^{2}}$, or
$\boldsymbol{(S^{1}\times S^{2})\;\\#\;L(p,q)}$._

Suppose $m$ and $R$ are disjoint nonseparating simple closed curves on the
boundary of a genus two handlebody $H$, $H[R]$ is the exterior of a knot in
$S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$, and $m$
represents the meridian of $H[R]$. In addition, suppose $m_{1}$ and $m_{2}$,
are the two distinguished meridian representatives of $H[R]$, obtained by
surgery on $R$ along the distinguished wave $\omega$ based at $R$, and suppose
$\\{D_{A},D_{B}\\}$ is any complete set of meridian disks for $H$, such that
the Heegaard diagram $\mathcal{D}$ of $R$ with respect to $\\{D_{A},D_{B}\\}$
is connected and has no cut-vertex. Finally, suppose $m$ is not isotopic to
$m_{1}$ or $m_{2}$ in $\partial H$.

Then there is a subarc $\omega^{\prime}$ of $\omega$ which is a wave based at
$m$ in $\mathcal{D}$. And then Lemma _8.1_ implies that the Heegaard diagram
of $m$ with respect to $\\{D_{A},D_{B}\\}$ is connected and has no cut-vertex.
It follows $\omega^{\prime}$ is a distinguished wave based at $m$, and $R$ is
obtained from $m$ by surgery on $m$ along $\omega^{\prime}$. This means the
complexity of the Heegaard diagram of $m$ with respect to $\\{D_{A},D_{B}\\}$
is greater than the complexity of the Heegaard diagram of $R$ with respect to
$\\{D_{A},D_{B}\\}$, which in turn is greater than the complexities of the
Heegaard diagrams of either $m_{1}$ or $m_{2}$ with respect to
$\\{D_{A},D_{B}\\}$.

###### Proof.

By Theorem 4.1, the distinguished wave $\omega$ based at $R$ depends only on
$R$ and $H$, and $\omega$ appears as a wave based at $R$ in any connected
Heegaard diagram $\mathcal{D}$ without cut-vertices of $R$ on $\partial H$. If
$m$ is any meridian representative of $H[R]$ in $\partial H$, disjoint from
$R$, then $m$ must have algebraic intersection number zero with $\omega$. So,
if $m$ is not disjoint from $\omega$, there exists a subarc $\omega^{\prime}$
of $\omega$ which appears in $\mathcal{D}$ as a wave based at $m$. Now Lemma
8.1 shows that, because $m$ is not isotopic to $m_{1}$, or $m_{2}$, the
Heegaard diagram of $m$ with respect to $\\{D_{A},D_{B}\\}$ is connected and
has no cut-vertex. By Theorem 5.2, $\omega^{\prime}$ is the distinguished wave
based at $m$, and $R$ is obtained from $m$ by surgery on $m$ along
$\omega^{\prime}$. This implies the complexities of the Heegaard diagrams of
$m$, $R$, $m_{1}$ and $m_{2}$, with respect to $\\{D_{A},D_{B}\\}$ are related
as claimed. ∎

###  8.1. The “shortest” meridian representative of a tunnel of a tunnel-
number-one knot in $\boldsymbol{S^{3}}$, $\boldsymbol{S^{1}\times S^{2}}$, or
$\boldsymbol{(S^{1}\times S^{2})\;\\#\;L(p,q)}$.

When at least one of the two distinguished meridian representatives $m_{1}$,
$m_{2}$ in $\partial H$ of a tunnel of a tunnel-number-one knot $k$ in a
3-manifold $M$ homeomorphic to $S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times
S^{2})\;\\#\;L(p,q)$ is not primitive or a proper-power in $H$, more can be
said. Namely the “shortest” meridian representative of $k$ exists and is well-
defined.

For, if one of $m_{1}$, $m_{2}$, say $m_{1}$, is not primitive or a proper-
power in $H$, then there exists a complete set $\Delta$ of meridian disks of
$H$ and Heegaard diagram $\mathcal{D}$ of $m_{1}$ and $m_{2}$ with respect to
$\Delta$ such that the subdiagram of $\mathcal{D}$ consisting of the edges of
$m_{1}$ in $\mathcal{D}$ is connected and has no cut-vertex. So $\mathcal{D}$
itself is connected, has no cut-vertex, and is a Heegaard diagram of $M$. Then
Theorem 7.1 applies and shows that there is a distinguished wave $\omega$
based at one of $m_{1}$, $m_{2}$ in $\mathcal{D}$. Finally, the results of
Section 6 show that $\omega$ appears in each Heegaard diagram of $m_{1}$ and
$m_{2}$ which is connected and has no cut-vertex. This justifies the following
definition.

######  Definition 8.4.

The “shortest” meridian representative of a tunnel of a tunnel-number-one knot
in $\boldsymbol{S^{3}}$, $\boldsymbol{S^{1}\times S^{2}}$, or
$\boldsymbol{(S^{1}\times S^{2})\;\\#\;L(p,q)}$.

Suppose $R$ is a nonseparating simple closed curve on the boundary of a genus
two handlebody, $H[R]$ has incompressible boundary, $H[R]$ is the exterior of
a knot in a 3-manifold $M$ homeomorphic to $S^{3}$, $S^{1}\times S^{2}$, or
$(S^{1}\times S^{2})\;\\#\;L(p,q)$, $\omega$ is the distinguished wave based
at $R$, and curves $m_{1}$ and $m_{2}$, obtained by surgery on $R$ along
$\omega$, are the two distinguished meridian representatives in $\partial H$
of the meridian of $H[R]$ in $M$.

If at least one of $m_{1},m_{2}$ is not primitive or a proper-power in $H$,
there exists a connected Heegaard diagram $\mathcal{D}$, without cut-vertices,
of $m_{1}$ and $m_{2}$ on $\partial H$, such that a distinguished wave
$\omega^{\prime}$, based at one of $m_{1},m_{2}$, appears in $\mathcal{D}$.
Then the _shortest_ meridian representative of $H[R]$ is the member of
$\\{m_{1},m_{2}\\}$ at which $\omega^{\prime}$ is not based.

##  9\. Unknotting tunnels and canonical constituent knots of theta curves in
$S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$.

Suppose $M$ is a 3-manifold homeomorphic to one of $(S^{1}\times
S^{2})\;\\#\;L(p,q)$, $S^{3}$ or $S^{1}\times S^{2}$. If $k$ is a knot in $M$,
an unknotting tunnel for $k$ is an arc $\tau$ in $M$ meeting $k$ only in its
endpoints such that the exterior of $k\,\cup\,\tau$ in $M$ is a genus two
handlebody. When $\tau$ is an unknotting tunnel for a knot $k$, $k\cup\tau$ is
a _$\theta$ curve_ in $M$, a regular neighborhood of $k\cup\tau$ in $M$ is a
genus two handlebody $H^{\prime}$, and the exterior of $k\cup\tau$ is also a
genus two handlebody $H$ such that $H$ and $H^{\prime}$ together form a genus
two Heegaard splitting of $M$ in which the exterior of $k$ in $M$ is obtained
by gluing the co-core of $\tau$ to $\partial H$ along a nonseparating simple
closed curve $R$ in $\partial H$.

The endpoints of $\tau$ on $k$ cut $k$ into a pair of arcs, say $\alpha_{1}$
and $\alpha_{2}$, in which case, $\tau\cup\alpha_{1}$ and $\tau\cup\alpha_{2}$
are said to be _constituent knots_ $k_{1}$ and $k_{2}$ of the $\theta$ curve
$k\cup\tau$. However, in general, constituent knots $k_{1}$ and $k_{2}$ of a
$\theta$ curve are not well defined, as isotopies of $\tau$ which slide the
endpoints of $\tau$ around on $k$ and perhaps pass each other on $k$ can
change $k_{1}$ and $k_{2}$.

If $H[R]$ has incompressible boundary, i.e. $R$ has essential intersections
with each meridional disk in $H$, the following procedure can be used to find
a pair of canonical constituent knots $k_{1}$ and $k_{2}$ of the $\theta$
curve $k_{\tau}$ such that $k_{1}$ and $k_{2}$ depend only on $k$ and $\tau$.

  1. (1)

Locate $R$ on $\partial H$.

  2. (2)

Locate a complete set, say $\\{D_{A},D_{B}\\}$, of meridian disks for $H$ such
that the Heegaard diagram $\mathcal{D}$ of $R$ with respect to
$\\{D_{A},D_{B}\\}$ is connected and has no cut-vertex.

  3. (3)

Locate the distinguished wave $\omega$ based at $R$ in $\mathcal{D}$.

  4. (4)

Let $N(R\cup\omega)$ be a regular neighborhood of $R\cup\omega$ in $\partial
H$, and let $m_{1}$ and $m_{2}$ be the two boundary components of $N$ not
isotopic to $R$ in $\partial H$.

  5. (5)

Then $m_{1}$ and $m_{2}$ are representatives of the meridian of $H[R]$ in $M$.
So $m_{1}$ and $m_{2}$ bound meridian disks $D_{m_{1}}$ and $D_{m_{2}}$ of
$H^{\prime}$ such that $H^{\prime}$ cut open along $D_{m_{1}}$ is a solid
torus $T_{2}$ with meridional disk $D_{m_{2}}$, and $H^{\prime}$ cut open
along $D_{m_{2}}$ is a solid torus $T_{1}$ with meridional disk $D_{m_{1}}$.
Then the core curves $k_{1}$ of $T_{1}$, transverse to $D_{m_{1}}$ at a point,
and $k_{2}$ of $T_{2}$, transverse to $D_{m_{2}}$ at a point, are a canonical
pair of constituent knots of the $\theta$ curve $k\cup\tau$ since the pair
$(k_{1},k_{2})$ depends only on $k$ and $\tau$.

######  Corollary 9.1.

An unknotting tunnel $\tau$ is a _(1,1)_ tunnel of $k$ if and only if one of
the canonical constituent knots of the $\theta$ curve $k\cup\tau$ is
unknotted.

###### Proof.

This follows directly from Theorem 10.1. ∎

######  Remark 9.2.

Corollary 9.1 is not true if the restriction to canonical constituent knots is
removed. Compare [MS91] [Proposition (1.3)], which is wrong because the
restriction to canonical constituent knots is omitted.

##  10\. Recognizing (1,1) tunnels of (1,1) knots in $S^{3}$ or $S^{1}\times
S^{2}$.

Suppose $M$ is a manifold homeomorphic to $S^{3}$ or $S^{1}\times S^{2}$. A
knot $k$ in $M$ is in (1,1) position if $k$ intersects each solid torus $V$,
$V^{\prime}$ of a genus one Heegaard splitting of $M$ in a single unknotted
arc. A knot in $M$ which has a (1,1) position is a (1,1) knot. If $k$ is in
(1,1) position, then $k$ has an unknotting tunnel $t_{V}$, respectively
$t_{V^{\prime}}$, such that $t_{V}$, respectively $t_{V^{\prime}}$, is a core
curve of $V$, respectively $V^{\prime}$. Such unknotting tunnels $t_{V}$ and
$t_{V^{\prime}}$ of $k$ form a _dual pair_ of (1,1) unknotting tunnels of $k$.

A (1,1) knot may have unknotting tunnels which are not (1,1) tunnels. A tunnel
of a tunnel-number-one knot $k$ which is not a (1,1) tunnel of $k$ is a
_regular_ tunnel of $k$.

If $k$ is in (1,1) position, drilling the unknotted arc $k\,\cap\,V$ out of
$V$, or the unknotted arc $k\cap V^{\prime}$ out of $V^{\prime}$, converts
$V$, respectively $V^{\prime}$, into a genus two handlebody $H$, respectively
$H^{\prime}$, in which the boundary of the co-core of each drilled out arc is
a meridian of $k$ that is primitive in $H$, respectively
$\pi_{1}(H^{\prime})$.

The following theorem provides a way to recognize that a tunnel is a (1,1)
tunnel of a (1,1) knot.

######  Theorem 10.1.

_(1,1) tunnel recognition._

Suppose $R$ is a nonseparating simple closed curve in the boundary of a genus
two handlebody $H$ such that $H[R]$ has incompressible boundary and $H[R]$ is
the exterior of a knot $k$ in a 3-manifold $M$ homeomorphic to $S^{3}$ or
$S^{1}\times S^{2}$. In addition, let $m_{1}$ and $m_{2}$, be the two simple
closed curve distinguished meridian representatives obtained from $R$ by
surgery on $R$ along the distinguished wave $\omega$ based at $R$ in $\partial
H$. Then $R$ is the boundary of a co-core of a _(1,1)_ tunnel for $k$, if and
only if one of $m_{1}$, $m_{2}$ is primitive in $H$.

###### Proof.

Suppose one of $m_{1}$, $m_{2}$, say $m_{1}$, is primitive in $H$. Then
filling $H[R]$ along $m_{1}$ yields a genus two Heegaard splitting
$(\Sigma;H,H^{\prime})$ of $M$ in which $m_{1}$ and $R$ bounding a complete
set of meridian disks of the complementary genus two handlebody $H^{\prime}$.
Furthermore, since $m_{1}$ is primitive in $H$, there exists a meridian disk
$D$ of $H$ such that $m_{1}\cap\,D$ is a single point of transverse
intersection. Then the meridian disk bounded by $m_{1}$ in $H^{\prime}$ and
the disk $D$ in $H$ form a trivial handle pair in $(\Sigma;H,H^{\prime})$. So
$(\Sigma;H,H^{\prime})$ destabilizes to a genus one Heegaard splitting
$(\Sigma^{\prime};V,V^{\prime})$ of $M$.

Next, the knot $k$ can be isotoped so it lies in $\Sigma$, is disjoint from
$R$, and has $k\cap\,m_{1}$ a single point of transverse intersection. Then
$k$ is clearly a (1,1) knot in $(\Sigma^{\prime};V,V^{\prime})$, and there
exists an unknotting tunnel $\tau$ for $k$ in $\Sigma$ such that $\tau\cap R$
is a single point of transverse intersection. Since the meridian disk of
$H^{\prime}$ bounded by $R$ becomes the meridian disk of $V^{\prime}$, $\tau$
becomes a core curve in $V^{\prime}$. So $\tau$ is a (1,1) unknotting tunnel
for $k$. This proves one direction of Theorem 10.1.

For the other direction of Theorem 10.1, suppose $k$ is in (1,1) position with
respect to a genus one Heegaard splitting $(\Sigma^{\prime};V,V^{\prime})$ of
$M$, and $\tau$ is a (1,1) tunnel of $k$ in $(\Sigma^{\prime};V,V^{\prime})$.
Then $k\,\cap\,V$, respectively $k\,\cap\,V^{\prime}$, is an unknotted arc in
$V$, respectively $V^{\prime}$. So drilling the unknotted arc $k\,\cap\,V$ out
of $V$ turns $V$ into a genus two handlebody $H$ in a genus two stabilization
$(\Sigma;H,H^{\prime})$ of $(\Sigma^{\prime};V,V^{\prime})$.

When $(\Sigma^{\prime};V,V^{\prime})$ is stabilized to
$(\Sigma;H,H^{\prime})$, the curve $R$ in $\Sigma^{\prime}$, bounding the
meridian disk of $V^{\prime}$, appears in $\Sigma$ as a curve bounding a
meridian disk of $H^{\prime}$. And, since the arc $k\,\cap\,V^{\prime}$ is
unknotted in $V^{\prime}$, $k\,\cap\,V^{\prime}$ can be isotoped, keeping its
endpoints fixed, into $\Sigma^{\prime}$ missing $R$. Then in
$(\Sigma;H,H^{\prime})$, $k$ and $R$ are disjoint curves in $\partial H$, the
boundary $m$ of the co-core of the drilled out arc $k\,\cap\,V$ lies in
$\partial H$, $m$ is disjoint from $R$, $m$ is primitive in $H$, and $m$ meets
$k$ in a single point of transverse intersection.

Now Lemma 8.1 shows that if $m$ is primitive in $H$, then $m$ must be isotopic
in $\partial H$ to one of the two distinguished meridian representatives
$m_{1}$ and $m_{2}$ of $H[R]$ obtained by surgery on $R$ along the
distinguished wave $\omega$ based at $R$. So either $m_{1}$ or $m_{2}$ must be
primitive in $H$. ∎

##  11\. Recognizing genus two Heegaard diagrams of $S^{3}$, $S^{1}\times
S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$.

The following procedure uses distinguished waves and shortest meridian
representatives, along with the major results of [HOT80] and [NO85] that every
connected genus two Heegaard diagram of $S^{3}$, $S^{1}\times S^{2}$, or
$(S^{1}\times S^{2})\;\\#\;L(p,q)$ has a wave, to determine if a genus two
Heegaard diagram of a closed 3-manifold is a Heegaard diagram of $S^{3}$,
$(S^{1}\times S^{2})\;\\#\;L(p,q)$ or $S^{1}\times S^{2}$.

###  11.1. A procedure to recognized genus two Heegaard diagrams of
$\boldsymbol{S^{3}}$, $\boldsymbol{S^{1}\times S^{2}}$, or
$\boldsymbol{(S^{1}\times S^{2})\;\\#\;L(p,q)}$

Suppose $R_{1}$ and $R_{2}$ are disjoint, nonparallel, nonseparating simple
closed curves on the boundary of a genus two handlebody $H$, and let $M$ be
the closed 3-manifold obtained by first adding 2-handles to $\partial H$ along
$R_{1}$ and $R_{2}$ to obtain $H[R_{1},R_{2}]$, and then filling the 2-sphere
boundary of $H[R_{1},R_{2}]$ with a 3-ball. Then, leaving aside the case in
which $R_{1}$ or $R_{2}$ bounds a disk in $H$, the following procedure can be
used to determine if $M$ is homeomorphic to $S^{3}$, $S^{1}\times S^{2}$, or
$(S^{1}\times S^{2})\;\\#\;L(p,q)$.

  1. (1)

Check, using say Theorem 12.5, whether $R_{1}$ or $R_{2}$ is primitive or a
proper-power in $H$.

  2. (2)

If $R_{1}$ or $R_{2}$ is primitive or a proper-power in $H$, then:

    1. (a)

If $H_{1}(M,\mathbb{Z})\neq 0,\,\mathbb{Z}$ or
$\mathbb{Z}\oplus\mathbb{Z}/p,\;p>1$, $M$ is not homeomorphic to $S^{3}$,
$S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$. Stop!

    2. (b)

If $H_{1}(M,\mathbb{Z})=0$, $M\simeq S^{3}$;

    3. (c)

If $H_{1}(M,\mathbb{Z})=\mathbb{Z}$, $M\simeq S^{1}\times S^{2}$;

    4. (d)

If $H_{1}(M,\mathbb{Z})=\mathbb{Z}\oplus\mathbb{Z}/p,\;p>1$,
$M\simeq(S^{1}\times S^{2})\;\\#\;L(p,q)$.

    5. (e)

Stop!

  3. (3)

Locate a complete set $\Delta$ of meridional disks of $H$ such that the
Heegaard diagram $\mathcal{D}$ of $\\{R_{1},R_{2}\\}$ with respect to $\Delta$
is connected and has no cut-vertex.

  4. (4)

Determine if there is a wave $\omega$ in $\mathcal{D}$ based at $R_{1}$ or
$R_{2}$.

  5. (5)

If no wave based at $R_{1}$ or $R_{2}$ in $\mathcal{D}$ exists, $M$ is not
homeomorphic to $S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times
S^{2})\;\\#\;L(p,q)$. Stop!

  6. (6)

If there is a wave $\omega$ based at $R_{1}$ or $R_{2}$ in $\mathcal{D}$, and
$\omega$ is based at $R_{1}$, respectively $R_{2}$, take $R_{2}$, respectively
$R_{1}$, to be the shortest member of $\\{R_{1},R_{2}\\}$ in $\mathcal{D}$.

  7. (7)

If, say $R_{1}$, is the shortest member of $\\{R_{1},R_{2}\\}$, look for a
complete set of meridian disks $\Delta^{\prime}$ of $H$ such that the Heegaard
diagram $\mathcal{D}^{\prime}$ of $R_{1}$ with respect to $\Delta^{\prime}$ is
connected and has no cut-vertex.

  8. (8)

Locate the distinguished wave $\omega^{\prime}$ based at $R_{1}$ in
$\mathcal{D}^{\prime}$.

  9. (9)

If $R_{2}$ does not have algebraic intersection number zero with
$\omega^{\prime}$ in $\mathcal{D}^{\prime}$, $R_{2}$ does not represent the
meridian of $H[R_{1}]$ and $\mathcal{D}^{\prime}$ is not a Heegaard diagram of
$S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$. Stop!

  10. (10)

If $R_{2}$ has algebraic intersection number zero with $\omega^{\prime}$ in
$\mathcal{D}^{\prime}$, perform surgery on $R_{1}$ along $\omega^{\prime}$ to
obtain the distinguished meridian representatives $m_{1}$ and $m_{2}$ of
$H[R_{1}]$. Replace $\\{R_{1},R_{2}\\}$ with $\\{m_{1},m_{2}\\}$, and go to
(1) above.

######  Theorem 11.1.

Procedure _11.1_ correctly determines if genus two Heegaard diagrams are
Heegaard diagrams of $S^{3}$, $S^{1}\times S^{2}$ or $(S^{1}\times
S^{2})\;\\#\;L(p,q)$.

###### Proof.

Most of the proof that Procedure 11.1 correctly recognizes genus two Heegaard
diagram of $S^{3}$, $S^{1}\times S^{2}$ or $(S^{1}\times S^{2})\;\\#\;L(p,q)$
follows directly from the major results of [HOT80] and [NO85] that every
connected genus two Heegaard diagram of $S^{3}$, $S^{1}\times S^{2}$ or
$(S^{1}\times S^{2})\;\\#\;L(p,q)$ has a wave. However, we mention two points.

(1) Continuing with the notation of Procedure 11.1, observe that if
$\omega^{\prime}$ is the distinguished wave based at $R_{1}$ in Step (8) of
Procedure 11 and $R_{2}$ intersects $\omega$ algebraically zero times in Step
(9) of Procedure 11.1, then the pair of curves $(R_{1},R_{2})$ and the pair of
curves $(m_{1},m_{2})$ have the same normal closure in $\pi_{1}(\partial H)$.
It follows that $R_{1}$ and $R_{2}$ are the curves of a genus two Heegaard
diagram of $M$ if and only if $m_{1}$ and $m_{2}$ are the curves of a genus
two Heegaard diagram of $M$. Furthermore, the Heegaard diagram of
$\\{m_{1},m_{2}\\}$ with respect to the complete set of meridian disks
$\Delta^{\prime}$ for $H$ in Step (7) of Procedure 11.1 has smaller complexity
than the Heegaard diagram of $\\{R_{1},R_{2}\\}$ with respect to the complete
set of meridian disks $\Delta^{\prime}$ for $H$.

(2) If $R_{2}$ does not intersect $\omega^{\prime}$ algebraically zero times
in Step (9) of Procedure 11.1, Lemma 5.1 together with Lemma 5.3 imply that
the Heegaard diagram $\mathcal{D}^{\prime}$ of $\\{R_{1},R_{2}\\}$ on
$\partial H$ in Step (7) of Procedure 11.1 is not a Heegaard diagram of
$S^{3}$, $S^{1}\times S^{2}$ or $(S^{1}\times S^{2})\;\\#\;L(p,q)$. ∎

##  12\. Computing the depth of a tunnel of a tunnel-number-one knot in
$S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$.

Suppose $M$ is a manifold homeomorphic to one of $(S^{1}\times
S^{2})\;\\#\;L(p,q)$, $S^{3}$ or $S^{1}\times S^{2}$, and suppose $R$ is a
nonseparating simple closed curve on the boundary of a genus two handlebody
$H$ such that the manifold $H[R]$, obtained by adding a 2-handle to $H$ along
$R$, is the exterior of a knot $k$ in $M$. In addition, let $\tau$ be the
unknotting tunnel of $k$ dual to $R$ on $\partial H$. In [CM1]–[CM9] Cho and
McCullogh described procedures for computing the “depth” of an unknotting
tunnel of a knot in $S^{3}$, along with procedures for computing sets of
“rational invariants” of unknotting tunnels of knots in $S^{3}$. Later, in
[I11], Ishihara adapted these procedures to compute the depth and rational
invariants of knots in $S^{3}$ using Heegaard diagrams of $S^{3}$. The
procedure described in 12.1 below uses distinguished waves in genus two
Heegaard diagrams of $M$ to compute the depth of tunnel-number-one knots in
$M$, and so generalizes the depth computation procedures of Cho, McCullogh and
Ishihara from $S^{3}$ to $S^{1}\times S^{2}$ and $(S^{1}\times
S^{2})\;\\#\;L(p,q)$.

######  Definition 12.1.

Unknotting paths of an unknotting tunnel.

Suppose $R$ is a nonseparating simple closed curve on the boundary of a genus
two handlebody $H$ such that the manifold $H[R]$, obtained by adding a
2-handle to $H$ along $R$, is the exterior of a knot $k$ in a manifold $M$
homeomorphic to $S^{3}$, $S^{1}\times S^{2}$, or $(S^{1}\times
S^{2})\;\\#\;L(p,q)$. Then $R$ determines an unknotting tunnel $\tau$ for $k$,
and there is an _unknotting path_ of $\tau$ consisting of a finite sequence
$R_{i}$, $0\leq i\leq n$, of simple closed curves on $\partial H$, starting
with $R_{0}$ = $R$ and ending with $R_{n}$ primitive or a proper-power in $H$,
such that for $0\leq i<n$, $R_{i}$ and $R_{i+1}$ are disjoint and $R_{i+1}$
represents the meridian of $H[R_{i}]$ in $M$.

######  Definition 12.2.

Depth of an unknotting tunnel.

The _depth_ of a tunnel $\tau$ of a tunnel-number-one knot $k$ in $S^{3}$,
$S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$ is the minimal
value of $n$ that occurs over all unknotting paths of $\tau$.

###  12.1. A procedure to compute the depth of an unknotting tunnel.

Suppose $R$ is a nonseparating simple closed curve on the boundary of a genus
two handlebody $H$ such that the manifold $H[R]$, obtained by adding a
2-handle to $H$ along $R$, is the exterior of a knot $k$ in $S^{3}$,
$S^{1}\times S^{2}$, or $(S^{1}\times S^{2})\;\\#\;L(p,q)$, and $\tau$ is the
unknotting tunnel of $k$ dual to $R$ on $\partial H$. Then the following
procedure computes the depth of $\tau$ by producing a well-defined unique
minimal length sequence $\mathcal{P}=\\{R_{i}\\}$, $0\leq i\leq n$, of simple
closed curves on $\partial H$, starting with $R_{0}$ = $R$ and ending with
$R_{n}$ primitive or a proper-power in $H$, such that for $0\leq i<n$, $R_{i}$
and $R_{i+1}$ are disjoint and $R_{i+1}$ represents the meridian of
$H[R_{i}]$.

  1. (1)

Set depth = 0.

  2. (2)

Determine if $R$ is primitive or a proper-power in $H$, using, say, Theorem
12.5.

  3. (3)

If $R$ is primitive or a proper-power in $H$, stop.

  4. (4)

Locate a complete set $\\{D_{A},D_{B}\\}$ of meridional disks of $H$ such that
the Heegaard diagram $\mathcal{D}$ of $R$ with respect to $\\{D_{A},D_{B}\\}$
is connected and has no cut-vertex.

  5. (5)

Locate the distinguished wave $\omega$ based at $R$ in $\mathcal{D}$, and
perform surgery on $R$ along $\omega$ to obtain the distinguished meridian
representatives $m_{1}$ and $m_{2}$ of $H[R]$.

  6. (6)

If the Heegaard diagram $\mathcal{D}^{\prime}$ of $\\{m_{1},m_{2}\\}$ with
respect to $\\{D_{A},D_{B}\\}$ is not connected, both $m_{1}$ and $m_{2}$ are
primitive or proper-powers in $H$. In this case, increment depth by 1, replace
$R$ with either $m_{1}$ or $m_{2}$, and go to step (2) above.

  7. (7)

Otherwise, determine which member of $\\{m_{1},m_{2}\\}$ is shortest, replace
$R$ with the shortest member of $\\{m_{1},m_{2}\\}$, increment depth by 1, and
go to step (2) above.

Note that each time the current curve $R$ is replaced in Procedure 12.1, it is
replaced by a curve of smaller complexity. So Procedure 12.1 terminates in a
finite number of steps.

Next, Lemma 12.4 shows that minimal length unknotting paths always pass from a
curve $c$ to one of the two distinguished meridian representatives of $H[c]$.
This implies there exists a finite graph $G$ such that $G$ carries all
possible minimal length unknotting paths, as per Definition 12.2, of a given
initial curve. Then analysis of the minimal length unknotting paths in $G$
will establish Theorem 12.3 below.

######  Theorem 12.3.

Procedure _12.1_ correctly computes depth.

######  Lemma 12.4.

Minimal length unknotting paths always pass from a curve $c$ to one of the two
distinguished meridian representatives of $H[c]$.

###### Proof.

Suppose, to the contrary, that there exists a minimal length unknotting path,
say $\mathcal{P}$, which at some point does not pass from a curve $c$ to one
of the two distinguished meridian representatives of $H[c]$.

Then, consider Figure 11, and suppose Figure 11a has the following properties:

  1. (1)

Vertices $\mathrm{A}$ and $\mathrm{B}$ of Figure 11a are successive vertices
of $\mathcal{P}$.

  2. (2)

The curve $R_{B}$ in $\partial H$, represented by vertex $\mathrm{B}$ of
$\mathcal{P}$, is disjoint from the curve $R_{A}$ in $\partial H$, represented
by vertex $\mathrm{A}$ of $\mathcal{P}$, $R_{B}$ is a meridian of $H[R_{A}]$,
but $R_{B}$ is not one of the two distinguished meridian representatives of
$H[R_{A}]$.

  3. (3)

$\mathrm{A}$ and $\mathrm{B}$ are the last pair of successive vertices in
$\mathcal{P}$ which satisfy (2).

Lemma 8.1 shows that if $c$ is a curve disjoint from $R_{A}$, $c$ represents a
meridian of $H[R_{A}]$, and $c$ is primitive or a proper-power in $H$, then
$c$ is one of the two distinguished meridian representatives of $H[R_{A}]$. It
follows the curve $R_{B}$ is not primitive or a proper-power in $H$. Thus $B$
is not the terminal vertex of $\mathcal{P}$, and so $B$ is followed
immediately in $\mathcal{P}$ by a vertex $\mathrm{C}$.

Next, since (3) holds, $R_{C}$ must be one of the two distinguished meridian
representatives of $H[R_{B}]$. In addition, since $R_{A}$ and $R_{B}$ are
disjoint, and neither $R_{A}$ nor $R_{B}$ is primitive or a proper-power in
$H$, one of $R_{A}$, $R_{B}$ must have a distinguished wave missing the other.
Since $R_{B}$ is not one of the two distinguished meridian representatives of
$R_{A}$, there must be a distinguished wave based at $R_{B}$ missing $R_{A}$.
But then $R_{A}$ and $R_{C}$ are both distinguished meridian representatives
of $H[R_{B}]$. Now $R_{C}$ can’t be isotopic to $R_{A}$ in $\partial H$, since
otherwise $\mathcal{P}$ backtracks from $A\rightarrow B\rightarrow A$. So
$R_{A}$ and $R_{C}$ are disjoint, $R_{C}$ is a meridian representative of
$H[R_{A}]$, and $\mathcal{P}$ can be shortened by going directly from $A$ to
$C$, as suggested by Figure 11b. ∎

Figure 11. Minimal length unknotting paths of a tunnel-number-one knot in
$\boldsymbol{S^{3}}$, $\boldsymbol{S^{1}\times S^{2}}$, or
$\boldsymbol{(S^{1}\times S^{2})\;\\#\;L(p,q)}$ always pass from a curve
$\boldsymbol{c}$ to a curve which is one of the two distinguished meridian
representatives of $\boldsymbol{H[c]}$.

The following result of Cohen, Metzler, and Zimmerman makes it possible to
determine if a cyclically reduced word in a free group of rank two is
primitive or a proper-power of a primitive.

######  Theorem 12.5 (CMZ81).

Suppose a cyclic conjugate of

$w=A^{m_{1}}B^{n_{1}}\dots A^{m_{j}}B^{n_{j}}$

is a member of a basis of $F(A,B)$, where $j\geq 1$ and each indicated
exponent is nonzero. Then, after perhaps replacing $A$ by $A^{-1}$ or $B$ by
$B^{-1}$, there exists $e>0$ such that:

$m_{1}=\dots=m_{j}=1,\quad\text{and}\quad\\{n_{1},\dots,n_{j}\\}=\\{e,e+1\\},$

or

$\\{m_{1},\dots,m_{j}\\}=\\{e,e+1\\},\quad\text{and}\quad
n_{1}=\dots=n_{j}=1.$

Note that if $w$ in $F(A,B)$ has the form $w=AB^{n_{1}}\dots AB^{n_{j}}$, say,
with $j\geq 1$ and $\\{n_{1},\dots,n_{j}\\}=\\{e,e+1\\}$, then the
automorphism $A\mapsto AB^{-e}$ of $F(A,B)$ reduces the length of $w$, so
repeated applications of such automorphisms can be used to determine if a
given word $w$ in $F(A,B)$ is a primitive.

###  12.2. A graph carrying all minimal length unknotting paths.

Lemma 12.4 shows that minimal length unknotting paths always pass from a curve
$c$ to one of the two distinguished meridian representatives of $H[c]$. This
implies there exists a finite graph $G$ with the property that $G$ carries all
possible minimal length unknotting paths, as per Definition 12.2, of a given
initial curve.

The graph $G$ can be produced by starting with an initial vertex representing
the initial curve whose minimal length unknotting paths are being sought, and
applying the following procedure recursively to add new vertices and edges to
$G$ until the procedure fails to add new vertices to $G$.

  1. (1)

If $V$ is a vertex of $G$, there are no directed edges of $G$ leaving $V$, and
the curve $R_{V}$ in $\partial H$ represented by $V$, is not primitive in $H$,
then:

    1. (a)

Let $m_{1}$ and $m_{2}$ be the two distinguished meridian representatives of
$H[R_{V}]$ obtained from $R_{V}$ by performing surgery on $R_{V}$ along the
distinguished wave based at $R_{V}$ in $\partial H$, and if $m_{1}$,
respectively $m_{2}$, is not represented by a vertex of $G$, add a vertex,
$V_{1}$ representing $m_{1}$, respectively, $V_{2}$ representing $m_{2}$, to
$G$.

    2. (b)

Add a directed edge to $G$ which leaves $V$ and connects $V$ to the vertex of
$G$ representing $m_{1}$. Similarly, add a directed edge to $G$ which leaves
$V$ and connects $V$ to the vertex of $G$ representing $m_{2}$.

  2. (2)

If $V$ is a vertex of $G$, there are no directed edges of $G$ leaving $V$, and
the curve $R_{V}$ in $\partial H$ represented by $V$ is primitive in $H$, do
nothing.

###  12.3. A subgraph $\boldsymbol{G^{*}}$ of $\boldsymbol{G}$ which carries
the unknotting path described in Subsection 12.1

Subsection 12.1 describes a procedure for computing a particular unknotting
path $\mathcal{P}$ of a given initial curve. This subsection describes a
subgraph $G^{*}$ of $G$ whose vertices are the simple closed curves in
$\partial H$ examined while computing $\mathcal{P}$.

The unknotting path $\mathcal{P}$, as described in 12.1, is a sequence
$\mathcal{P}=\\{R_{i}\\}$, $0\leq i\leq n$, of simple closed curves on
$\partial H$ which starts with an initial curve $R_{0}$ and ends with a curve
$R_{n}$, primitive or a proper-power in $H$, such that for $0\leq i<n$,
$R_{i}$ and $R_{i+1}$ are disjoint, and at each step with $0\leq i<n$,
$R_{i+1}$ is obtained from $R_{i}$ by choosing one of the two distinguished
meridian representatives of $H[R_{i}]$ in $\partial H$ as $R_{i+1}$.

Given $\mathcal{P}$, let the vertices of $G^{*}$ be the vertices of $G$ which
represent $R_{0}$ and both distinguished meridian representatives of each
$R_{i}$ in $\mathcal{P}$, for $0\leq i<n$. In addition, if $V_{i}$, $0\leq
i<n$, is a vertex of $G^{*}$ which represents a curve in $\mathcal{P}$, add
the pair of directed edges of $G$ leaving $V_{i}$ to the set of directed edges
of $G^{*}$.

Next, recall that, when $R_{i}$, $0\leq i<n$, is a curve in $\mathcal{P}$, one
member of the pair of distinguished meridian representatives of $H[R_{i}]$
lies in $\mathcal{P}$, the other member of the pair of distinguished meridian
representatives of $H[R_{i}]$ does not lie in $\mathcal{P}$. In particular, if
$m_{i,1}$ and $m_{i,2}$, are the two distinguished meridian representatives of
$H[R_{i}]$, $m_{i,1}\in\mathcal{P}$, $m_{i,2}\notin\mathcal{P}$, $m_{i,2}$ is
not primitive or a proper-power in $H$, $V_{i,1}$ is the vertex of $G^{*}$
representing $m_{i,1}$, and $V_{i,2}$ is the vertex of $G^{*}$ representing
$m_{i,2}$, add a directed edge to $G^{*}$ pointing from $V_{i,2}$ to
$V_{i,1}$.

###  12.4. Drawing $\boldsymbol{G}$ And its subgraph $\boldsymbol{G^{*}}$.

The graphs $G^{*}$ and $G$ have relatively simple structures which become
apparent when $G^{*}$ and $G$ are drawn. Since $G^{*}$ is planar, we begin by
describing a planar embedding of $G^{*}$, and then indicate how to add edges
and perhaps vertices to the embedding of $G^{*}$ to obtain $G$.

The vertices of $G^{*}$ and $G$ can be embedded in the plane so that they lie
in two disjoint vertical columns, with the vertices of $G^{*}$ representing
curves in $\mathcal{P}$ lying in the first column, and all other vertices of
$G^{*}$ and $G$ lying in the second column. To wit, suppose $\mathrm{X}$,
$\mathrm{Y}$ and $\mathrm{Y^{\prime}}$ are three vertices of $G^{*}$ such that
$\mathrm{X}$ and $\mathrm{Y}$ represent successive members of $\mathcal{P}$,
the curve $\mathrm{Y^{\prime}}$ represents is not in $\mathcal{P}$, the curve
$R_{X}$ in $\partial H$ that $\mathrm{X}$ represents is not primitive or a
proper-power in $H$, and the curves $R_{Y}$ and $R_{Y^{\prime}}$ in $\partial
H$ that $\mathrm{Y}$ and $\mathrm{Y^{\prime}}$ represent are the two
distinguished meridians of $H[R_{X}]$. Then embed $\mathrm{X}$ and
$\mathrm{Y}$ in the first column with $\mathrm{Y}$ directly below
$\mathrm{X}$, and embed $\mathrm{Y^{\prime}}$ in the second column at a height
between the heights of $\mathrm{X}$ and $\mathrm{Y}$. Then add appropriate
directed edges of $G^{*}$, after which a generic local embedded subgraph of
$G^{*}$ should have the form of Figure 12a.

With $G^{*}$ embedded, the edges and vertices of $G$ which do not belong to
$G^{*}$ can be added to the embedding of $G^{*}$. These edges and vertices of
$G$ are the edges and vertices of $G$ which can only be traversed or reached
via a directed path from the initial vertex of $G^{*}$ that passes through a
“primed” vertex of $G^{*}$ which represents a curve in $\partial H$ that is
not primitive or a proper-power in $H$.

For example, consider vertex $\mathrm{B^{\prime}}$ of Figure 12, and let $R$
be the curve in $\partial H$ which $\mathrm{B^{\prime}}$ represents. Since
there is a directed edge of $G^{*}$ leaving vertex $\mathrm{B^{\prime}}$, $R$
is not primitive or a proper-power in $H$, and so surgery on $R$ along the
distinguished wave based at $R$ in $\partial H$ yields two distinguished
meridian representatives, say $m_{1}$ and $m_{2}$, of $H[R]$ in $\partial H$.
Furthermore, since the directed edge of $G^{*}$ leaving vertex
$\mathrm{B^{\prime}}$ in $G^{*}$ ends at vertex $\mathrm{B}$ of $G^{*}$,
vertex $\mathrm{B}$ of $G^{*}$ represents one of $m_{1}$, $m_{2}$. So, without
loss, let vertex $\mathrm{B}$ of $G^{*}$ represent $m_{1}$. Then one of the
following occurs:

  1. (1)

Both $m_{1}$ and $m_{2}$ are primitives or proper-powers in $H$.
In this case Figure 15b illustrates the situation. In Figure 15, vertices of
$G$ representing curves in $\partial H$ that are primitive or proper-powers in
$H$ are shown as black squares, vertex $\mathrm{C^{\prime}}$ of Figure 15b
represents the curve $m_{2}$, and there are no directed edges of $G$ leaving
either vertex $\mathrm{B}$ or vertex $\mathrm{C^{\prime}}$ of $G$. Hence no
vertices or edges of $G$ are embedded in either column below $\mathrm{B}$ or
$\mathrm{C^{\prime}}$.

  2. (2)

The distinguished wave based at $m_{1}$ in $\partial H$ is disjoint from
$m_{2}$.
In this case, vertex $\mathrm{B}$ represents a curve, say $S$ in $\partial H$,
which is not primitive or a proper-power in $H$, so surgery on $S$ along the
distinguished wave based at $S$ in $\partial H$ yields two distinguished
meridian representatives, say $m_{S1}$ and $m_{S2}$ of $H[S]$ in $\partial H$,
one of which must be isotopic to $m_{2}$ in $\partial H$. Thus $m_{2}$ is
represented either by vertex $\mathrm{C}$ or vertex $\mathrm{C^{\prime}}$ of
$G^{*}$, and there is a directed edge of $G$ connecting vertex
$\mathrm{B^{\prime}}$ to either $\mathrm{C}$ or vertex $\mathrm{C^{\prime}}$
of $G^{*}$. Figures 13a and 13b depict these two possibilities.

  3. (3)

The distinguished wave based at $m_{2}$ in $\partial H$ is disjoint from
$m_{1}$.
In this case, there may exist several vertices of $G$ which are only
accessible via directed paths in $G$ that start at the initial vertex of $G$
and pass through vertex $\mathrm{B^{\prime}}$ of $G^{*}$. The simplest case
occurs when there is only one such vertex, say $\mathrm{E^{\prime}}$. In which
case, $\mathrm{E^{\prime}}$ represents the curve $m_{2}$ in $\partial H$.

Then there must be two directed edges of $G$ leaving $\mathrm{E^{\prime}}$,
one of which must connect $\mathrm{E^{\prime}}$ to vertex $\mathrm{B}$ of $G$,
while the second edge of $G$ leaving $\mathrm{E^{\prime}}$ must connect
$\mathrm{E^{\prime}}$ to either vertex $\mathrm{C}$ or vertex
$\mathrm{C^{\prime}}$ of $G^{*}$, as indicated by Figures 13c and 13d. (Note
the curve in $\partial H$ which vertex $\mathrm{B}$ represents is not
primitive or a proper-power in $H$ in this case, since otherwise there exists
more than one vertex of $G$ which is only accessible via directed paths in $G$
that start at the initial vertex of $G$ and pass through vertex
$\mathrm{B^{\prime}}$ of $G^{*}$. Hence there are directed edges of $G^{*}$
connecting vertex $\mathrm{B}$ of $G^{*}$ to vertices $\mathrm{C}$ and
$\mathrm{C^{\prime}}$ of $G^{*}$.)

In the general case, there is more than one vertex of $G$ which is only
accessible via directed paths in $G$ that start at the initial vertex of $G$
and pass through vertex $\mathrm{B^{\prime}}$ of $G^{*}$.

In this case, there exists a sequence of pairs of disjoint curves
$(m_{i,1},m_{i,2})$, $0\leq i\leq n$, in $\partial H$ with each $m_{i,1}$
isotopic to $m_{1}$ in $\partial H$, for $0\leq i\leq n$, and $m_{0,2}$ =
$m_{2}$. Furthermore, no $m_{i,2}$ is primitive or a proper-power in $H$ for
$0\leq i<n$, and there is a distinguished wave $\omega_{i}$ based at each
$m_{i,2}$ in $\partial H$, for $0\leq i<n$, with $\omega_{i}$ disjoint from
$m_{i,1}$, such that $m_{i+1,1}$ and $m_{i+1,2}$ are the two distinguished
meridian representatives of $H[m_{i,2}]$ obtained by surgery on $m_{i,2}$
along $\omega_{i}$.

We mention that each curve $m_{i+1,2}$, $0\leq i<n$, can be obtained by
surgery on $m_{i+1,2}$ along $\omega_{i}$ in a Heegaard diagram of $m_{i,2}$
that has minimal complexity. It follows $i$ reaches a value $n$ such that
either $m_{n,2}$ is primitive or a proper-power in $H$, or $m_{n,2}$ is not
primitive or a proper-power in $H$, but the distinguished wave $\omega_{n}$
based at $m_{n,2}$ in $\partial H$ is not disjoint from $m_{n,1}$.

Suppose $\mathrm{V_{i}^{\prime}}$ is the vertex of $G$ representing $m_{i,2}$
for $0\leq i\leq n$. Then, for $0\leq i<n$, one of the two directed edges of
$G$ leaving $\mathrm{V_{i}^{\prime}}$ connects $\mathrm{V_{i}^{\prime}}$ to
vertex $\mathrm{B}$ of $G$, while the other directed edge leaving
$\mathrm{V_{i}^{\prime}}$ connects $\mathrm{V_{i}^{\prime}}$ to
$\mathrm{V_{i+1}^{\prime}}$.

The location of $\mathrm{V_{n}^{\prime}}$ in $G$ depends upon whether the
curve $m_{1}$, which vertex $\mathrm{B}$ represents in $G^{*}$, is primitive
or a proper-power in $H$, and upon whether the curve $m_{n,2}$, which vertex
$\mathrm{V_{n}^{\prime}}$ represents in $G$, is primitive or a proper-power in
$H$. In particular, there are the following cases:

    1. (a)

The curve $m_{1}$ is primitive in $H$.
In this case, the subgraph of $G$ only accessible via vertex
$\mathrm{B^{\prime}}$ of $G^{*}$, must appear as in Figure 15c with vertex
$\mathrm{V_{n-1,2}}$ of $G$ appearing as vertex $\mathrm{E^{\prime}}$ in
Figure 15c, and vertex $\mathrm{V_{n,2}}$ of $G$ appearing as vertex
$\mathrm{C^{\prime}}$ in Figure 15c.

    2. (b)

The curve $m_{1}$ is not primitive or a proper-power in $H$.
In this case, the subgraph of $G$ only accessible via vertex
$\mathrm{B^{\prime}}$ of $G^{*}$, must appear as in Figure 13c or Figure 13d
with vertex $\mathrm{V_{n-1,2}}$ of $G$ appearing as vertex
$\mathrm{E^{\prime}}$ in Figure 13c or Figure 13d, and vertex
$\mathrm{V_{n,2}}$ of $G$ appearing as vertex $\mathrm{C^{\prime}}$ in Figure
13c or as vertex $\mathrm{C}$ in Figure 13d, depending upon whether the curve
$m_{n,2}$ is isotopic in $\partial H$ to the curve which vertex
$\mathrm{C^{\prime}}$ of $G^{*}$ represents or to the curve which vertex
$\mathrm{C}$ of $G^{*}$ represents.

Figure 12. Representative local generic subgraphs of $\boldsymbol{G^{*}}$.
(Figure 12b is meant to indicate that there are additional edges and perhaps
additional vertices of $G$ accessible from the initial vertex of $G^{*}$ only
via directed paths in $G^{*}$ passing through a “primed” vertex of $G^{*}$,
e.g. vertex $\mathrm{B^{\prime}}$ or vertex $\mathrm{C^{\prime}}$, of Figure
12a.) Figure 13. Four types of subgraphs of $\boldsymbol{G}$. The four types
of subgraphs of $G$ which can arise from the process of adding edges to Figure
12b that are only accessible from vertex $\mathrm{B^{\prime}}$ in Figure 12b
when the curve which vertex $\mathrm{B}$ represents in $\partial H$ is not
primitive or a proper-power in $H$. (There may be several descendants of
$\mathrm{B^{\prime}}$ which appear as intervening vertices on the line from
vertex $\mathrm{B^{\prime}}$ to vertex $\mathrm{E^{\prime}}$, as suggested by
the dotted lines in Figure 13c and Figure 13d.)

###  12.5. Minimal length directed paths in $\boldsymbol{G}$.

######  Definition 12.6.

Minimal directed path length in $\boldsymbol{G}$.

If $\mathrm{V}$ is a vertex of $G$, and $P$ is a directed path in $G$ from the
initial vertex of $G$ to $\mathrm{V}$, let $L(P)$ be the number of edges of
$G$ which occur in $P$. Then let $L(\mathrm{V})$ be the minimum value of
$L(P)$ over the set of all directed paths in $G$ from the initial vertex of
$G$ to $\mathrm{V}$.

######  Lemma 12.7.

_Minimal path length equalities in $\boldsymbol{G}$._

Suppose $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are three successive
vertices of $G^{*}$ representing three successive curves lying in the
particular unknotting path $\mathcal{P}$ described in Subsection _12.1_. In
addition, suppose $\mathrm{B^{\prime}}$ and $\mathrm{C^{\prime}}$ are vertices
of $G^{*}$ such that the two directed edges of $G^{*}$ leaving vertex
$\mathrm{A}$ point to vertices $\mathrm{B}$ and $\mathrm{B^{\prime}}$, while
the two directed edges of $G^{*}$ leaving vertex $\mathrm{B}$ point to
vertices $\mathrm{C}$ and $\mathrm{C^{\prime}}$. Then
$L(\mathrm{B})=L(\mathrm{B^{\prime}})$ implies
$L(\mathrm{C})=L(\mathrm{C^{\prime}})$.

###### Proof.

First, consider Figure 14a. Then
$L(\mathrm{C})=\text{min}\\{L(\mathrm{B})+1,L(\mathrm{C^{\prime}})+1\\}$, and
$L(\mathrm{C^{\prime}})=\text{min}\\{L(\mathrm{B})+1,L(\mathrm{B^{\prime}})+1\\}$.
By hypothesis, $L(\mathrm{B})=L(\mathrm{B^{\prime}})$. So we have
$L(\mathrm{C^{\prime}})=L(\mathrm{B})+1$, $L(\mathrm{C})=L(\mathrm{B})+1$, and
$L(\mathrm{C})=L(\mathrm{C^{\prime}})$. Next, observe that
$L(\mathrm{C})=L(\mathrm{B})+1$, and $L(\mathrm{C})=L(\mathrm{C^{\prime}})$
still hold when $\mathrm{B^{\prime}}$ and $\mathrm{C^{\prime}}$ are connected
by a path in $G$ containing vertices of $G$.

Second, consider Figure 14b. Then
$L(\mathrm{C})=\text{min}\\{L(\mathrm{B})+1,L(\mathrm{B^{\prime}})+1,L(\mathrm{C^{\prime}})+1\\}$,
and $L(\mathrm{C^{\prime}})=L(\mathrm{B})+1$. By hypothesis,
$L(\mathrm{B})=L(\mathrm{B^{\prime}})$. So
$L(\mathrm{C^{\prime}})=L(\mathrm{B})+1$, $L(\mathrm{C})=L(\mathrm{B})+1$, and
$L(\mathrm{C})=L(\mathrm{C^{\prime}})$. Next, observe that
$L(\mathrm{C})=L(\mathrm{B})+1$, and $L(\mathrm{C})=L(\mathrm{C^{\prime}})$
still hold when $\mathrm{B^{\prime}}$ and $\mathrm{C}$ are connected by a path
in $G$ containing vertices of $G$.

Finally, note the conclusions above still hold when:

  1. (1)

There are directed edges of $G$ ending at $\mathrm{C}$ which originate at
vertices of $G$, distinct from $\mathrm{C^{\prime}}$, that are only accessible
from the initial vertex of $G$ via directed paths in $G$ which pass through
$\mathrm{C^{\prime}}$.

  2. (2)

There is no edge of $G$ connecting $\mathrm{C^{\prime}}$ and $\mathrm{C}$.

∎

Figure 14. Two graphs used in showing that
$\mathbf{L(B)\boldsymbol{=}L(B^{\prime})}$ implies
$\mathbf{L(C)\boldsymbol{=}L(C^{\prime})}$. Figure 15. Local directed paths in
$\boldsymbol{G}$ which terminate with vertices—shown as filled
rectangles—representing curves in $\boldsymbol{\partial H}$ that are
primitives or proper-powers in $\boldsymbol{H}$.

###### Proof of Theorem _12.3_.

It follows readily from the forms of $G^{*}$ and $G$ as detailed above, that
there is only one ‘unprimed’ vertex of $G$ which represents a curve in
$\partial H$ that is primitive or a proper-power in $H$, and only one ‘primed’
vertex of $G$ which represents a curve in $\partial H$ that is primitive or a
proper-power in $H$, and that these two vertices must arise in $G$ in one of
the three configurations illustrated in Figure 15. Then Lemma 12.7, applied
inductively, shows that $L(\mathrm{B})=L(\mathrm{B^{\prime}})$ in each
subfigure of Figure 15. It follow that, in all cases, Procedure 12.1 produces
a minimal length unknotting path for a given curve. ∎

##  13\. Other waves based at a fixed curve.

The results of the previous sections show that if $R$ is a nonseparating
simple closed curve on the boundary of a genus two handlebody $H$ such that
$H[R]$ has incompressible boundary, then there exists a distinguished wave
$\omega$, based at $R$ in $\partial H$, such that $\omega$ appears in each
connected Heegaard diagram $\mathcal{D}$, without cut-vertices, of $R$ on