#  A Note on Sectional Curvatures of Hermitian Manifolds∗ 00footnotetext:
∗Correspondence<EMAIL_ADDRESS>
Hongjun Li1∗ and Pandeng Cao2
1School of Mathematics and Statistics, Henan University, Kaifeng 475004, China
Email<EMAIL_ADDRESS>
2School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
<EMAIL_ADDRESS>
Abstract First, we derive expression of the Chern sectional curvature of a
Hermitian manifold in local complex coordinates. As an application, we find
that a Hermitian metric is Kähler if the Riemann sectional curvature and the
Chern sectional curvature coincide. Second, we prove that the sectional
curvature restricted to orthogonal 2-planes of a G-Kähler-like manifold with
non-negative (resp. non-positive) sectional curvature can take its maximum
(resp. minimum) at a holomorphic plane section. And we also prove that the
holomorphic bisectional curvature of a Kähler-like manifold with non-negative
(resp. non-positive) Chern sectional curvature can take its maximum (resp.
minimum) at the holomorphic sectional curvature.

Key words Chern sectional curvature, G-Kähler-like manifold, Kähler-like
manifold.

2020 MR Subject Classification: 53C55, 53B35

##  1 Introduction

Suppose $(M,h)$ is an $n$-dimensional Hermitian manifold, and $g={\rm Re\,}h$
is the background Riemannian metric associated to $h$. Let us denote by $D$
and $\nabla$ the Chern (or Hermitian) connection and the Levi-Civita (or
Riemannian) connection, respectively. The curvature tensors of the Chern
connection and the Levi-Civita connection are denoted by ${\mathfrak{R}}$ and
$R$. It is well known that $h$ is Kähler if and only if the Chern connection
$D$ coincide with the Levi-Civita $\nabla$ (refer to [2, 7, 10], etc.). Hence
${\mathfrak{R}}$ is the linear extension of $R$ over ${\mathbb{C}}$ under
Kähler hypothesis.

Let $z=(z^{1},\cdots,z^{n})$ and $x=(x^{1},\cdots,x^{2n})$ be local complex
and real coordinates of a point $p\in M$, where
$z^{\alpha}=x^{\alpha}+\sqrt{-1}x^{n+\alpha}$, $1\leq\alpha\leq n$. In local
coordinates, we denote by
$h=h_{\alpha\bar{\beta}}(z)dz^{\alpha}d\bar{z}^{\beta}$ and $g={\rm
Re\,}{h}=g_{ij}(x){d}x^{i}{d}x^{j}$. In this paper, we assume that lowercase
Greek indices run from 1 to $n$ and lowercase Latin indices run from 1 to
$2n$. We define a bundle isomorphism ${}_{o}:TM\rightarrow T^{1,0}M$ by

$u_{o}=\frac{1}{2}(u-\sqrt{-1}Ju),\quad\forall u\in TM,$

where $J$ is the complex structure on $M$. Let $u,~{}v\in T_{p}M$, we set
$\xi=u_{o},~{}\eta=v_{o}\in T^{1,0}_{p}M$. If $h$ is Kähler, then

$R(J{u},u,v,Jv)=2{\mathfrak{R}}\left(\xi,\bar{\xi},{\eta},\bar{\eta}\right),$
(1.1)
$R(u,v,v,u)=\displaystyle\frac{1}{2}\left[{\mathfrak{R}}\left(\xi,\bar{\eta},{\eta},\bar{\xi}\right)+{\mathfrak{R}}\left({\eta},\bar{\xi},\xi,\bar{\eta}\right)-{\mathfrak{R}}\left(\xi,\bar{\eta},\xi,\bar{\eta}\right)-{\mathfrak{R}}\left(\eta,\bar{\xi},{\eta},\bar{\xi}\right)\right].$
(1.2)

The first formula can be referred to [10], and the second formula can be
referred to [6, 8]. Especially, if we take $v=u$ in (1.1) or take $v=J{u}$ in
(1.2), then (1.1) and (1.2) become

$R(Ju,u,u,Ju)=2{\mathfrak{R}}(\xi,\bar{\xi},\xi,\bar{\xi})$ (1.3)

under Kähler hypothesis.

In this paper, we will give a geometric interpretation of the right side in
(1.2) when $h$ is non-Kähler. In local coordinates, the right side in (1.2)
can be written as

$\displaystyle\frac{1}{2}{\mathfrak{R}}_{\alpha\bar{\beta}\gamma\bar{\delta}}\big{(}\xi^{\alpha}\bar{\eta}^{\beta}-{\eta}^{\alpha}\bar{\xi}^{\beta}\big{)}\big{(}{\eta}^{\gamma}\bar{\xi}^{\delta}-\xi^{\gamma}\bar{\eta}^{\delta}\big{)},$
(1.4)

where
${\mathfrak{R}}_{\alpha\bar{\beta}\gamma\bar{\delta}}={\mathfrak{R}}\left(\displaystyle\frac{\partial}{\partial
z^{\alpha}},\frac{\partial}{\partial\bar{z}^{\beta}},\frac{\partial}{\partial
z^{\gamma}},\frac{\partial}{\partial\bar{z}^{\delta}}\right)$. We find that

$\displaystyle\frac{\displaystyle\frac{1}{2}{\mathfrak{R}}_{\alpha\bar{\beta}\gamma\bar{\delta}}\big{(}\xi^{\alpha}\bar{\eta}^{\beta}-{\eta}^{\alpha}\bar{\xi}^{\beta}\big{)}\big{(}{\eta}^{\gamma}\bar{\xi}^{\delta}-\xi^{\gamma}\bar{\eta}^{\delta}\big{)}}{\left(h_{\alpha\bar{\beta}}\xi^{\alpha}\bar{\xi}^{\beta}\right)\cdot\left(h_{\gamma\bar{\delta}}\eta^{\gamma}\bar{\eta}^{\delta}\right)-\displaystyle\frac{1}{4}\left[h_{\alpha\bar{\beta}}\left(\xi^{\alpha}\bar{\eta}^{\beta}+\eta^{\alpha}\bar{\xi}^{\beta}\right)\right]^{2}}$
(1.5)

is just the sectional curvature of the metric connection induced by the Chern
connection, we call it the Chern sectional curvature of $(M,h)$.

######  Theorem 1.1.

Let $(M,h)$ be a Hermitian manifold with the background Riemannian metric
$g={\rm Re\,}h$. Suppose $D$ is the metric connection on the Riemannian
manifold $(M,g)$ induced by the Chern connection. For arbitrary
$u=u^{i}\frac{\partial}{\partial x^{i}},~{}v=v^{i}\frac{\partial}{\partial
x^{i}}\in T_{p}M$, we have

$g\left((D^{2}u)(v,u),v\right)=\displaystyle\frac{1}{2}{\mathfrak{R}}_{\alpha\bar{\beta}\gamma\bar{\delta}}\big{(}\xi^{\alpha}\bar{\eta}^{\beta}-{\eta}^{\alpha}\bar{\xi}^{\beta}\big{)}\big{(}{\eta}^{\gamma}\bar{\xi}^{\delta}-\xi^{\gamma}\bar{\eta}^{\delta}\big{)},$
(1.6)

where $\xi=u_{o}$ and $\eta=v_{o}$. Especially,

$g\left((D^{2}u)(Ju,u),Ju\right)={2}{\mathfrak{R}}_{\alpha\bar{\beta}\gamma\bar{\delta}}\xi^{\alpha}\bar{\xi}^{\beta}\xi^{\gamma}\bar{\xi}^{\delta}.$
(1.7)

From (1.6) in Theorem 1.1, we can see

$g\left((D^{2}u)(v,u),v\right)=g\left((D^{2}v)(u,v),u\right).$

Recently, the first author and Qiu [3] proved that a Hermitian manifold such
that (1.3) is Kähler. Hence (1.7) in Theorem 1.1 can imply the following
corollary.

######  Corollary 1.2.

Let $(M,h)$ be a Hermitian manifold such that the Riemann sectional curvature
and the Chern sectional curvature coincide. Then $h$ is a Kähler metric.

Does the sectional curvature restricted to orthogonal 2-planes of a Kähler
manifold always take its maximum or minimum at some holomorphic plane section?
According to Lu's result in [5], we have a positive answer if the Kähler
manifold has non-negative or non-positive sectional curvature. Next, we will
generalize Lu's result on G-Kähler-like manifolds.

######  Theorem 1.3.

Let $(M,h)$ be a G-Kähler-like manifold with with non-negative (resp. non-
positive) sectional curvature. Then the sectional curvature restricted to
orthogonal 2-planes of $(M,h)$ can take its maximum (resp. minimum) at some
holomorphic plane section.

Moreover, we have

######  Theorem 1.4.

Let $(M,h)$ be a Kähler-like manifold with non-negative (resp. non-positive)
Chern sectional curvature. Then the holomorphic bisectional curvature of
$(M,h)$ can take its maximum (resp. minimum) at the holomorphic sectional
curvature.

G-Kähler-like manifolds and Kähler-like manifolds are introduced by Yang and
Zheng [9], which are classes of non-Kähler manifolds. We extend the Riemannian
curvature tensor $R=(R_{ijkl})$ linearly over ${\mathbb{C}}$. For a general
Hermitian metric, Gray [1] discovered
$R_{\alpha\beta\gamma\delta}=R_{\bar{\alpha}\bar{\beta}\bar{\gamma}\bar{\delta}}=0$,
but the types $R_{\alpha\beta\gamma\bar{\delta}}$ and
$R_{\alpha\beta\bar{\gamma}\bar{\delta}}$ may not vanish.

######  Definition 1.5.

[9] A Hermitian metric $h$ is called Kähler-like, if its holomorphic sectional
curvature tensor satisfies
$\mathfrak{R}_{\alpha\bar{\beta}\gamma\bar{\delta}}=\mathfrak{R}_{\gamma\bar{\beta}\alpha\bar{\delta}}$
for all $\alpha,\,\beta,\,\gamma,\,\delta=1,2,\cdots,n$. Similarly, $h$ is
called Gray-Kähler-like, or G-Kähler-like for short, if
$R_{\alpha{\beta}\gamma\bar{\delta}}=R_{\alpha{\beta}\bar{\gamma}\bar{\delta}}=0$
for all $\alpha,\,\beta,\,\gamma,\,\delta=1,2,\cdots,n$.

The rest of this paper is arranged as follows. In Section 2, we introduce the
Chern sectional curvature and give proof of Theorem 1.1. In Section 3, we
provide a brief overview of complexified Riemannian curvature and give proofs
of Theorems 1.3 and 1.4.

##  2 The Chern sectional curvature

The Chern connection $D$ is the unique connection on the holomorphic tangent
bundle $T^{1,0}M$ which is compatible with the Hermitian metric $h$ and the
complex structure $J$. We denote by

$\theta=(\theta_{\beta}^{\alpha})=\left({\varGamma}_{\beta;\gamma}^{\alpha}dz^{\gamma}\right)$

the connection 1-form matrix of $D$, where
$\varGamma_{\beta;\gamma}^{\alpha}=h^{\bar{\lambda}\alpha}\displaystyle\frac{\partial
h_{\beta\bar{\lambda}}}{\partial z^{\gamma}}$ are connection coefficients of
$D$. The curvature tensor of $D$ is denoted by ${\mathfrak{R}}$ with
components

${\mathfrak{R}}_{\alpha\bar{\beta}\gamma\bar{\delta}}=-\displaystyle\frac{\partial^{2}h_{\alpha\bar{\beta}}}{\partial
z^{\gamma}\partial\bar{z}^{\delta}}+\displaystyle\frac{\partial
h_{\alpha\bar{\lambda}}}{\partial
z^{\gamma}}h^{\bar{\lambda}\kappa}\displaystyle\frac{\partial
h_{\kappa\bar{\beta}}}{\partial\bar{z}^{\delta}}.$ (2.1)

We define the holomorphic bisectional curvature by

$B(\xi,\eta)=\displaystyle\frac{{\mathfrak{R}}\left(\xi,\bar{\xi},{\eta},\bar{\eta}\right)}{h\left(\xi,{\xi}\right)h\left({\eta},{\eta}\right)},\quad\forall\xi,\,\eta\in
T^{1,0}M.$ (2.2)

Ulteriorly, the holomorphic sectional curvature is defined by

$H(\xi)=B(\xi,\xi)=\displaystyle\frac{{\mathfrak{R}}\left(\xi,\bar{\xi},\xi,\bar{\xi}\right)}{h\left(\xi,{\xi}\right)^{2}},\quad\forall\xi\in
T^{1,0}M.$ (2.3)

######  Lemma 2.1.

[2] Under the bundle isomorphism o any complex connection on $T^{1,0}M$
induces a metric connection on the Riemannian manifold $(M,g)$.

When no confusion can rise, we still denote by $D$ the metric connection on
the Riemannian manifold $(M,g)$, which is induced by the Chern connection $D$.
Then

$\displaystyle D\frac{\partial}{\partial
x^{\alpha}}=\frac{1}{2}\left(\theta_{\alpha}^{\beta}+\bar{\theta}_{\alpha}^{\beta}\right)\frac{\partial}{\partial
x^{\beta}}-\frac{\sqrt{-1}}{2}\left(\theta_{\alpha}^{\beta}-\bar{\theta}_{\alpha}^{\beta}\right)\frac{\partial}{\partial
x^{\beta+n}},$ (2.4) $\displaystyle D\frac{\partial}{\partial
x^{\alpha+n}}=\frac{\sqrt{-1}}{2}\left(\theta_{\alpha}^{\beta}-\bar{\theta}_{\alpha}^{\beta}\right)\frac{\partial}{\partial
x^{\beta}}+\frac{1}{2}\left(\theta_{\alpha}^{\beta}+\bar{\theta}_{\alpha}^{\beta}\right)\frac{\partial}{\partial
x^{\beta+n}}.$

For a general Hermitian manifold $(M,h)$, the induced metric connection $D$ is
compatible with the Riemannian metric $g$, but not torsion free.

######  Definition 2.2.

Let $(M,h)$ be a Hermitian manifold. We call

$\begin{split}K_{D}(u,v)=\displaystyle\frac{g((D^{2}u)(v,u),v)}{g(u,u)g(v,v)-g(u,y)^{2}}\end{split}$
(2.5)

the Chern sectional curvature of the 2-plane $\Pi(u,v)$ spanned by two
linearly independent tangent vectors $u=u^{i}\frac{\partial}{\partial
x^{i}},\,v=v^{i}\frac{\partial}{\partial x^{i}}\in TM$.

Next we give expression of the Chern sectional curvature in local complex
coordinates.

Proof of Theorem 1.1 We denote by $D\frac{\partial}{\partial
x^{i}}=\tilde{\theta}_{i}^{j}\frac{\partial}{\partial x^{j}}$ and
$\tilde{\theta}=(\tilde{\theta}_{i}^{j})$. It follows (2.4) that

$\tilde{\theta}=F{\rm diag\,}\left\\{\theta,\bar{\theta}\right\\}F^{-1},$
(2.6)

where $F=\left(\begin{array}[]{cr}I&I\\\
\mathrm{i}I&-\mathrm{i}I\end{array}\right)$ and $I$ is the $n\times n$ unit
matrix. In order to simplify the calculation process, we introduce the
following notations. Denote by

$\frac{\partial}{\partial x}=\left(\frac{\partial}{\partial
x^{1}},\cdots,\frac{\partial}{\partial
x^{2n}}\right),\quad\frac{\partial}{\partial z}=\left(\frac{\partial}{\partial
z^{1}},\cdots,\frac{\partial}{\partial
z^{n}}\right),\quad\frac{\partial}{\partial\bar{z}}=\left(\frac{\partial}{\partial\bar{z}^{1}},\cdots,\frac{\partial}{\partial\bar{z}^{n}}\right),$
$\mathbf{U}=(u^{1},\cdots,u^{2n}),\quad\mathbf{\Xi}=(\xi^{1},\cdots,\xi^{n}).$

By a straightforward computation, we have

$\displaystyle D^{2}u=$ $\displaystyle
u^{j}\left(d\tilde{\theta}_{j}^{i}-\tilde{\theta}_{j}^{k}\wedge\tilde{\theta}_{k}^{i}\right)\otimes\frac{\partial}{\partial
x^{i}}$ $\displaystyle=$
$\displaystyle\mathbf{U}\left(d\tilde{\theta}-\tilde{\theta}\wedge\tilde{\theta}\right)\otimes\left(\frac{\partial}{\partial
x}\right)^{\rm t}$ $\displaystyle=$
$\displaystyle(\mathbf{\Xi},\overline{\mathbf{\Xi}}){\rm
diag\,}\left\\{d\theta-\theta\wedge\theta,d\bar{\theta}-\bar{\theta}\wedge\bar{\theta}\right\\}\otimes\left(\frac{\partial}{\partial
z},\frac{\partial}{\partial\bar{z}}\right)^{\rm t}$ $\displaystyle=$
$\displaystyle\mathbf{\Xi}\left(\bar{\partial}\theta\right)\otimes\left(\frac{\partial}{\partial
z}\right)^{\rm
t}+\overline{\mathbf{\Xi}}\left({\partial}\bar{\theta}\right)\otimes\left(\frac{\partial}{\partial\bar{z}}\right)^{\rm
t}$ $\displaystyle=$
$\displaystyle\xi^{\alpha}\left(\bar{\partial}\theta_{\alpha}^{\mu}\right)\otimes\frac{\partial}{\partial
z^{\mu}}+\bar{\xi}^{\beta}\left({\partial}\bar{\theta}_{\beta}^{\mu}\right)\otimes\frac{\partial}{\partial\bar{z}^{\mu}}$
$\displaystyle=$
$\displaystyle\xi^{\alpha}\left({\mathfrak{R}}^{\mu}_{\alpha\gamma\bar{\delta}}dz^{\gamma}\wedge
d\bar{z}^{\delta}\right)\otimes\frac{\partial}{\partial
z^{\mu}}-\bar{\xi}^{\beta}\left(\overline{{\mathfrak{R}}^{\mu}_{\beta\delta\bar{\gamma}}}dz^{\gamma}\wedge
d\bar{z}^{\delta}\right)\otimes\frac{\partial}{\partial\bar{z}^{\mu}}.$

Hence

$(D^{2}u)(v,u)=\xi^{\alpha}{\mathfrak{R}}^{\mu}_{\alpha\gamma\bar{\delta}}\big{(}{\eta}^{\gamma}\bar{\xi}^{\delta}-\xi^{\gamma}\bar{\eta}^{\delta}\big{)}\frac{\partial}{\partial
z^{\mu}}-\bar{\xi}^{\beta}\overline{{\mathfrak{R}}^{\mu}_{\beta\delta\bar{\gamma}}}\big{(}{\eta}^{\gamma}\bar{\xi}^{\delta}-\xi^{\gamma}\bar{\eta}^{\delta}\big{)}\frac{\partial}{\partial\bar{z}^{\mu}},$

and

$\displaystyle g\left((D^{2}u)(v,u),y\right)=$
$\displaystyle\frac{1}{2}\xi^{\alpha}{\mathfrak{R}}^{\mu}_{\alpha\gamma\bar{\delta}}\big{(}{\eta}^{\gamma}\bar{\xi}^{\delta}-\xi^{\gamma}\bar{\eta}^{\delta}\big{)}h_{\mu\bar{\beta}}\bar{\eta}^{\beta}-\frac{1}{2}\bar{\xi}^{\beta}\overline{{\mathfrak{R}}^{\mu}_{\beta\delta\bar{\gamma}}}\big{(}{\eta}^{\gamma}\bar{\xi}^{\delta}-\xi^{\gamma}\bar{\eta}^{\delta}\big{)}h_{\alpha\bar{\mu}}{w}^{\alpha}$
$\displaystyle=$
$\displaystyle\frac{1}{2}{\mathfrak{R}}_{\alpha\bar{\beta}\gamma\bar{\delta}}\xi^{\alpha}\bar{\eta}^{\beta}\big{(}{\eta}^{\gamma}\bar{\xi}^{\delta}-\xi^{\gamma}\bar{\eta}^{\delta}\big{)}-\frac{1}{2}{\mathfrak{R}}_{\alpha\bar{\beta}\gamma\bar{\delta}}{\eta}^{\alpha}\bar{\xi}^{\beta}\big{(}{\eta}^{\gamma}\bar{\xi}^{\delta}-\xi^{\gamma}\bar{\eta}^{\delta}\big{)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}{\mathfrak{R}}_{\alpha\bar{\beta}\gamma\bar{\delta}}\big{(}\xi^{\alpha}\bar{\eta}^{\beta}-{\eta}^{\alpha}\bar{\xi}^{\beta}\big{)}\big{(}{\eta}^{\gamma}\bar{\xi}^{\delta}-\xi^{\gamma}\bar{\eta}^{\delta}\big{)}.$

Note that

$\left(Ju\right)_{o}=\displaystyle\frac{1}{2}\left(Ju-\sqrt{-1}J^{2}u\right)=\displaystyle\frac{\sqrt{-1}}{2}\left(u-\sqrt{-1}Ju\right)=J(u_{o}).$

In order to prove (1.7), we only replace $v$ and $\eta$ in (1.6) with $Ju$ and
$\sqrt{-1}v$, respectively.∎

According to above theorem, we have

$K_{D}(u,v)=\displaystyle\frac{\displaystyle\frac{1}{2}{\mathfrak{R}}_{\alpha\bar{\beta}\gamma\bar{\delta}}\big{(}\xi^{\alpha}\bar{\eta}^{\beta}-{\eta}^{\alpha}\bar{\xi}^{\beta}\big{)}\big{(}{\eta}^{\gamma}\bar{\xi}^{\delta}-\xi^{\gamma}\bar{\eta}^{\delta}\big{)}}{\left(h_{\alpha\bar{\beta}}\xi^{\alpha}\bar{\xi}^{\beta}\right)\cdot\left(h_{\gamma\bar{\delta}}\eta^{\gamma}\bar{\eta}^{\delta}\right)-\displaystyle\frac{1}{4}\left[h_{\alpha\bar{\beta}}\left(\xi^{\alpha}\bar{\eta}^{\beta}+\eta^{\alpha}\bar{\xi}^{\beta}\right)\right]^{2}}.$
(2.7)

Thus $K_{D}(u,v)=K_{D}(v,u)$.

##  3 Sectional curvatures of Hermitian manifolds

The Levi-Civita connection $\nabla$ is the unique connection on the tangent
bundle $TM$, which is torsion free and compatible with the background
Riemannian metric $g$. The sectional curvature tensor $R$ is defined by

$R(X,Y,Z,W)=g\left(\left(\left[\nabla_{X},\nabla_{Y}\right]-\nabla_{\left[X,Y\right]}\right)Z,W\right),\quad\forall
X,\,Y,\,Z,\,W\in TM.$ (3.1)

In local coordinates, components of the sectional curvature tensor $R$ are

$\displaystyle R_{ijkl}=$
$\displaystyle\frac{1}{2}\left(\frac{\partial^{2}g_{ik}}{{\partial
x^{j}}{\partial x^{l}}}+\frac{\partial^{2}g_{jl}}{{\partial x^{i}}{\partial
x^{k}}}-\frac{\partial^{2}g_{il}}{{\partial x^{j}}{\partial
x^{k}}}-\frac{\partial^{2}g_{jk}}{{\partial x^{i}}{\partial x^{l}}}\right)$
(3.2) $\displaystyle+g^{st}\left([jl,s][ik,t]-[jk,s][il,t]\right),$

where

$[jk,s]=\displaystyle\frac{1}{2}\left(\displaystyle\frac{\partial
g_{js}}{\partial x^{k}}+\displaystyle\frac{\partial g_{ks}}{\partial
x^{j}}-\displaystyle\frac{\partial g_{jk}}{\partial x_{s}}\right).$ (3.3)

The sectional curvature of the 2-plane $\Pi(u,v)$ is defined by

$K(u,v)=\displaystyle\frac{R(u,v,v,u)}{g(u,u)g(v,v)-g(u,v)^{2}}.$ (3.4)

We call $\Pi(u,Ju)$ a holomorphic plane section [10].

If $(M,h)$ is a Kähler manifold, then the induced metric connection $D$ and
the Levi-Civita connection $\nabla$ coincide, thus $K_{D}(u,v)=K(u,v)$.

Now we extend the background Riemannian metric $g={\rm Re\,}h$ linearly over
${\mathbb{C}}$ to the complexified tangent bundle
$T_{{\mathbb{C}}}M=TM\otimes{\mathbb{C}}=T^{1,0}M\oplus T^{0,1}M$, then

$\displaystyle g\left(\frac{\partial}{\partial
z^{\alpha}},\frac{\partial}{\partial z^{\beta}}\right)$
$\displaystyle=g\left(\frac{\partial}{\partial\bar{z}^{\alpha}},\frac{\partial}{\partial\bar{z}^{\beta}}\right)=0,$
$\displaystyle g\left(\frac{\partial}{\partial
z^{\alpha}},\frac{\partial}{\partial\bar{z}^{\beta}}\right)$
$\displaystyle=\overline{g\left(\frac{\partial}{\partial\bar{z}^{\alpha}},\frac{\partial}{\partial{z}^{\beta}}\right)}=\frac{1}{2}h_{\alpha\bar{\beta}}.$

Hence

$h\left(\xi,\eta\right)=2g\left(\xi,\bar{\eta}\right),\quad\forall\xi,\,\eta\in
T^{1,0}M.$ (3.5)

For simplicity, we denote by $h_{\alpha\beta}=h_{\bar{\alpha}\bar{\beta}}=0$.

Now we extend the Levi-Civita connection $\nabla$ linearly over ${\mathbb{C}}$
to the complexified tangent bundle $T_{{\mathbb{C}}}M$. The complexified
Christoffel symbols are given by [4]

$\varGamma_{AB}^{C}=\displaystyle\frac{1}{2}h^{CD}\left(\displaystyle\frac{\partial
h_{AD}}{\partial z^{B}}+\displaystyle\frac{\partial h_{BD}}{\partial
z^{A}}-\displaystyle\frac{\partial h_{AB}}{\partial z^{D}}\right)$ (3.6)

where $A,\,B,\,C,\,D\in\\{1,\cdots,n,\bar{1},\cdots,\bar{n}\\}$ and
$z^{A}=z^{\alpha}$ if $A=\alpha$, ${z}^{A}=\bar{z}^{\alpha}$ if
$A=\bar{\alpha}$. That is,

$\varGamma_{\beta\gamma}^{\alpha}=\displaystyle\frac{1}{2}h^{\bar{\lambda}\alpha}\left(\displaystyle\frac{\partial
h_{\beta\bar{\lambda}}}{\partial z^{\gamma}}+\displaystyle\frac{\partial
h_{\gamma\bar{\lambda}}}{\partial
z^{\beta}}\right),\quad\varGamma_{\bar{\beta}\gamma}^{\alpha}=\displaystyle\frac{1}{2}h^{\bar{\lambda}\alpha}\left(\displaystyle\frac{\partial
h_{\gamma\bar{\lambda}}}{\partial\bar{z}^{\beta}}-\displaystyle\frac{\partial
h_{\gamma\bar{\beta}}}{\partial\bar{z}^{\lambda}}\right).$ (3.7)

Then the complexified Riemannian curvature tensor components are defined by
[4]

$\begin{split}R_{ABCD}=&2g\left(\left[\nabla_{\frac{\partial}{\partial
z^{A}}},\nabla_{\frac{\partial}{\partial
z^{B}}}\right]\frac{\partial}{\partial z^{C}},\frac{\partial}{\partial
z^{D}}\right)\\\ =&h\left(\left[\nabla_{\frac{\partial}{\partial
z^{A}}},\nabla_{\frac{\partial}{\partial
z^{B}}}\right]\frac{\partial}{\partial
z^{C}},\frac{\partial}{\partial\bar{z}^{D}}\right).\end{split}$ (3.8)

According to [4],

$R_{\alpha\bar{\beta}\mu}^{\gamma}=R_{\alpha\bar{\beta}\mu\bar{\nu}}h^{\bar{\nu}\gamma}=-\left(\frac{\partial\varGamma_{\alpha\mu}^{\gamma}}{\partial\bar{z}^{\beta}}-\frac{\partial\varGamma^{\gamma}_{\bar{\beta}\mu}}{\partial
z^{\alpha}}+\varGamma_{\alpha\mu}^{\kappa}\varGamma_{\bar{\beta}\kappa}^{\gamma}-\varGamma^{\kappa}_{\bar{\beta}\mu}\varGamma_{\alpha\kappa}^{\gamma}-\varGamma^{\bar{\lambda}}_{\bar{\beta}\mu}\varGamma_{\alpha\bar{\lambda}}^{\gamma}\right).$
(3.9)

A direct computation shows

$\displaystyle-\left(\frac{\partial\varGamma_{\alpha\mu}^{\gamma}}{\partial\bar{z}^{\beta}}+\varGamma_{\alpha\mu}^{\kappa}\varGamma_{\bar{\beta}\kappa}^{\gamma}\right)h_{\gamma\bar{\nu}}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\frac{\partial^{2}h_{\alpha\bar{\nu}}}{\partial
z^{\mu}\partial\bar{z}^{\beta}}+\frac{\partial^{2}h_{\mu\bar{\nu}}}{\partial
z^{\alpha}\partial\bar{z}^{\beta}}\right)+\frac{1}{4}\left(\frac{\partial
h_{\alpha\bar{\lambda}}}{\partial z^{\mu}}+\frac{\partial
h_{\mu\bar{\lambda}}}{\partial
z^{\alpha}}\right)h^{\bar{\lambda}\kappa}\left(\frac{\partial
h_{\kappa\bar{\nu}}}{\partial\bar{z}^{\beta}}+\frac{\partial
h_{\kappa\bar{\beta}}}{\partial\bar{z}^{\nu}}\right),$
$\displaystyle\left(\frac{\partial\varGamma^{\gamma}_{\bar{\beta}\mu}}{\partial
z^{\alpha}}+\varGamma^{\kappa}_{\bar{\beta}\mu}\varGamma_{\alpha\kappa}^{\gamma}\right)h_{\gamma\bar{\nu}}$
$\displaystyle=$
$\displaystyle\frac{\partial}{\partial{z}^{\alpha}}\left(\varGamma_{\bar{\beta}\mu}^{\gamma}h_{\gamma\bar{\nu}}\right)+\varGamma_{\bar{\beta}\mu}^{\kappa}\left(\varGamma_{\alpha\kappa}^{\gamma}h_{\gamma\bar{\nu}}-\frac{\partial
h_{\kappa\bar{\nu}}}{\partial{z}^{\alpha}}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{\partial^{2}h_{\mu\bar{\nu}}}{\partial
z^{\alpha}\partial\bar{z}^{\beta}}-\frac{\partial^{2}h_{\mu\bar{\beta}}}{\partial
z^{\alpha}\partial\bar{z}^{\nu}}\right)+\frac{1}{4}\left(\frac{\partial
h_{\mu\bar{\lambda}}}{\partial\bar{z}^{\beta}}-\frac{\partial
h_{\mu\bar{\beta}}}{\partial\bar{z}^{\lambda}}\right)h^{\bar{\lambda}\kappa}\left(\frac{\partial
h_{\alpha\bar{\nu}}}{\partial{z}^{\kappa}}-\frac{\partial
h_{\kappa\bar{\nu}}}{\partial{z}^{\alpha}}\right),$
$\displaystyle\varGamma^{\bar{\lambda}}_{\bar{\beta}\mu}\varGamma_{\alpha\bar{\lambda}}^{\gamma}h_{\gamma\bar{\nu}}=\frac{1}{4}h^{\bar{\lambda}\kappa}\left(\frac{\partial
h_{\kappa\bar{\beta}}}{\partial{z}^{\mu}}-\frac{\partial
h_{\mu\bar{\beta}}}{\partial{z}^{\kappa}}\right)\left(\frac{\partial
h_{\alpha\bar{\nu}}}{\partial\bar{z}^{\lambda}}-\frac{\partial
h_{\alpha\bar{\lambda}}}{\partial\bar{z}^{\nu}}\right).$

Hence

$\displaystyle R_{\alpha\bar{\beta}\mu\bar{\nu}}=$
$\displaystyle-\frac{1}{2}\left(\frac{\partial^{2}h_{\alpha\bar{\nu}}}{\partial
z^{\mu}\partial\bar{z}^{\beta}}+\frac{\partial^{2}h_{\mu\bar{\beta}}}{\partial
z^{\alpha}\partial\bar{z}^{\nu}}\right)$ (3.10)
$\displaystyle+\frac{1}{4}\left(\frac{\partial
h_{\alpha\bar{\lambda}}}{\partial z^{\mu}}+\frac{\partial
h_{\mu\bar{\lambda}}}{\partial
z^{\alpha}}\right)h^{\bar{\lambda}\kappa}\left(\frac{\partial
h_{\kappa\bar{\nu}}}{\partial\bar{z}^{\beta}}+\frac{\partial
h_{\kappa\bar{\beta}}}{\partial\bar{z}^{\nu}}\right)$
$\displaystyle-\frac{1}{4}\left(\frac{\partial
h_{\mu\bar{\lambda}}}{\partial\bar{z}^{\beta}}-\frac{\partial
h_{\mu\bar{\beta}}}{\partial\bar{z}^{\lambda}}\right)h^{\bar{\lambda}\kappa}\left(\frac{\partial
h_{\kappa\bar{\nu}}}{\partial{z}^{\alpha}}-\frac{\partial
h_{\alpha\bar{\nu}}}{\partial{z}^{\kappa}}\right)$
$\displaystyle-\frac{1}{4}\left(\frac{\partial
h_{\alpha\bar{\lambda}}}{\partial\bar{z}^{\nu}}-\frac{\partial
h_{\alpha\bar{\nu}}}{\partial\bar{z}^{\lambda}}\right)h^{\bar{\lambda}\kappa}\left(\frac{\partial
h_{\kappa\bar{\beta}}}{\partial{z}^{\mu}}-\frac{\partial
h_{\mu\bar{\beta}}}{\partial{z}^{\kappa}}\right).$

If $h$ is Kähler, then $R={\mathfrak{R}}$. Recently, Yang and Zheng [9] proved
$h$ is is Kähler if $R={\mathfrak{R}}$.

The complexified Riemannian curvature tensor $R_{ABCD}$ is also skew-symmetric
in the first two and last two entries, symmetric between the first two and
last two entries, and satisfies the Bianchi identity. Hence

$\displaystyle R\left(u,v,v,u\right)$ (3.11) $\displaystyle=$ $\displaystyle
2{\rm
Re\,}\left[R(\xi,\eta,\eta,\bar{\xi})+R(\xi,\eta,\bar{\eta},\xi)\right]+R(\xi,\bar{\eta},\eta,\bar{\xi})$
$\displaystyle-R(\xi,\eta,\bar{\xi},\bar{\eta})-\frac{1}{2}\left[R(\xi,\bar{\eta},\xi,\bar{\eta})+R(\eta,\bar{\xi},\eta,\bar{\xi})\right].$

Especially,

$\displaystyle R(u,Ju,Ju,u)=2R(\xi,\bar{\xi},\xi,\bar{\xi}).$ (3.12)

For a G-Kähler-like metric, Yang and Zheng [9] pointed out the vanishing of
$R_{\alpha\beta\bar{\gamma}\bar{\delta}}$ plus the first Bianchi identity
imply
$R_{\alpha\bar{\beta}\gamma\bar{\delta}}=R_{\gamma\bar{\beta}\alpha\bar{\delta}}$.
In order to prove Theorems 1.3 and 1.4, we need to introduce the following
theorem of Lu [5].

######  Theorem 3.1.

[5] Let $A_{\alpha\bar{\beta}\mu\bar{\nu}}$ be a set complex numbers
$(\alpha,\,\beta,\,\mu,\,\nu=1,\cdots,n)$ satisfying

  * (1)

$A_{\alpha\bar{\beta}\mu\bar{\nu}}=A_{\mu\bar{\beta}\alpha\bar{\nu}}=A_{\alpha\bar{\nu}\mu\bar{\beta}}$
and
$A_{\alpha\bar{\beta}\mu\bar{\nu}}=\overline{A_{\beta\bar{\alpha}\nu\bar{\mu}}}$,

  * (2)

$A_{\alpha\bar{\beta}\mu\bar{\nu}}\left(\xi^{\alpha}\bar{\eta}^{\beta}-{\eta}^{\alpha}\bar{\xi}^{\beta}\right)\overline{\left(\xi^{\nu}\bar{\eta}^{\mu}-{\eta}^{\nu}\bar{\xi}^{\mu}\right)}\geq
0$ (or $\leq 0$) for all complex numbers
$\xi^{\alpha},\,{\eta}^{\alpha}\,(\alpha=1,\cdots,n)$. Then we always have

$|A_{\alpha\bar{\beta}\mu\bar{\nu}}\xi^{\alpha}\bar{\xi}^{\beta}{\eta}^{\mu}\bar{\eta}^{\nu}|^{2}\leq
A_{\alpha\bar{\beta}\mu\bar{\nu}}\xi^{\alpha}\bar{\xi}^{\beta}\xi^{\mu}\bar{\xi}^{\nu}A_{\alpha\bar{\beta}\mu\bar{\nu}}{\eta}^{\alpha}\bar{\eta}^{\beta}{\eta}^{\mu}\bar{\eta}^{\nu}.$
(3.13)

Proof of Theorem 1.3 Yang and Zheng [9] pointed out
$R_{\alpha{\mu}\bar{\beta}\bar{\nu}}=0$ is equivalent to

$R_{\alpha\bar{\beta}\mu\bar{\nu}}=R_{\mu\bar{\beta}\alpha\bar{\nu}}\quad{\rm
or}\quad R_{\alpha\bar{\beta}\mu\bar{\nu}}=R_{\alpha\bar{\nu}\mu\bar{\beta}}.$
(3.14)

Hence the G-Kähler-like condition implies

$R_{\alpha\bar{\beta}\mu\bar{\nu}}=R_{\mu\bar{\beta}\alpha\bar{\nu}}=R_{\alpha\bar{\nu}\mu\bar{\beta}}=\overline{R_{\beta\bar{\alpha}\nu\bar{\mu}}},$
(3.15)
$\begin{split}R\left(u,v,v,u\right)=&R\left(\xi,\bar{\eta},\eta,\bar{\xi}\right)-\frac{1}{2}\left[R\left(\xi,\bar{\eta},\xi,\bar{\eta})+R(\eta,\bar{\xi},\eta,\bar{\xi}\right)\right]\\\
=&\displaystyle\frac{1}{2}R_{\alpha\bar{\beta}\mu\bar{\nu}}\big{(}\xi^{\alpha}\bar{\eta}^{\beta}-{\eta}^{\alpha}\bar{\xi}^{\beta}\big{)}\overline{\left(\xi^{\nu}\bar{\eta}^{\mu}-{\eta}^{\nu}\bar{\xi}^{\mu}\right)}.\end{split}$
(3.16)

If we replace $v$ and $\eta$ in (3.16) with $Jv$ and $\sqrt{-1}\eta$, then

$R\left(u,Jv,Jv,u\right)=R\left(\xi,\bar{\xi},\eta,\bar{\eta}\right)+\frac{1}{2}\left[R\left(\xi,\bar{\eta},\xi,\bar{\eta}\right)+R\left(\eta,\bar{\xi},\eta,\bar{\xi}\right)\right],$
(3.17)

and thus

$R\left(u,v,v,u\right)+R\left(u,Jv,Jv,u\right)=2R\left(\xi,\bar{\xi},\eta,\bar{\eta}\right).$
(3.18)

Moreover, the sectional curvature of the G-Kähler-like metric $h$ is non-
positive or non-negative, so is $R\left(\xi,\bar{\xi},\eta,\bar{\eta}\right)$.
It follows (3.15), (3.16) and Theorem 3.1 that

$\left|R\left(\xi,\bar{\xi},\eta,\bar{\eta}\right)\right|^{2}\leq
R\left(\xi,\bar{\xi},\xi,\bar{\xi}\right)R\left(\eta,\bar{\eta},\eta,\bar{\eta}\right).$
(3.19)

And the non-positive or non-negative sectional curvature condition plus (3.16)
and (3.17) imply

$\left|R\left(\xi,\bar{\eta},\xi,\bar{\eta}\right)+R\left(\eta,\bar{\xi},\eta,\bar{\xi}\right)\right|\leq
2R\left(\xi,\bar{\xi},\eta,\bar{\eta}\right),$

which with (3.16) yields

$R\left(u,v,v,u\right)^{2}\leq
4R\left(\xi,\bar{\xi},\eta,\bar{\eta}\right)^{2}\leq
4R\left(\xi,\bar{\xi},\xi,\bar{\xi}\right)R\left(\eta,\bar{\eta},\eta,\bar{\eta}\right)=R\left(Ju,u,u,Ju\right)R\left(Jv,v,v,Jv\right).$

by using (3.12) and (3.19). We have

$\left|R\left(u,v,v,u\right)\right|\leq{\rm
max\,}\left\\{\left|R\left(Ju,u,u,Ju\right)\right|,\left|R\left(Jv,v,v,Jv\right)\right|\right\\}.$
(3.20)

Note that $g(y,Jy)=0$ for all $y\in TM$. Hence when we assume $g(u,v)=0$, if
$h$ has non-negative sectional curvature then

$K\left(u,v\right)\leq\mathop{{\rm max\,}}\limits_{y\in
TM}K\left(y,Jy\right),$

and if $h$ has non-positive sectional curvature then

$K\left(u,v\right)\geq\mathop{{\rm min\,}}\limits_{y\in
TM}K\left(y,Jy\right).$

where $u,\,v,\,y$ are unit vectors. We complete the proof.∎

Proof of Theorem 1.4 The Kähler-like condition implies

${\mathfrak{R}}_{\alpha\bar{\beta}\mu\bar{\nu}}={\mathfrak{R}}_{\mu\bar{\beta}\alpha\bar{\nu}}={\mathfrak{R}}_{\alpha\bar{\nu}\mu\bar{\beta}}=\overline{{\mathfrak{R}}_{\beta\bar{\alpha}\nu\bar{\mu}}}.$
(3.21)

By (1.6) in Theorem 1.1, the non-negative or non-positive Chern sectional
curvature means

${\mathfrak{R}}_{\alpha\bar{\beta}\mu\bar{\nu}}\big{(}\xi^{\alpha}\bar{\eta}^{\beta}-{\eta}^{\alpha}\bar{\xi}^{\beta}\big{)}\overline{\left(\xi^{\nu}\bar{\eta}^{\mu}-{\eta}^{\nu}\bar{\xi}^{\mu}\right)}\geq
0~{}~{}{\rm or}~{}\leq 0.$ (3.22)

By Theorem 3.1, we have

$\left|{\mathfrak{R}}\left(\xi,\bar{\xi},\eta,\bar{\eta}\right)\right|^{2}\leq{\mathfrak{R}}\left(\xi,\bar{\xi},\xi,\bar{\xi}\right){\mathfrak{R}}\left(\eta,\bar{\eta},\eta,\bar{\eta}\right).$
(3.23)

Therefore, if $h$ has non-negative Chern sectional curvature then

${\mathfrak{R}}\left(\xi,\bar{\xi},\eta,\bar{\eta}\right)\leq\mathop{{\rm
max\,}}\limits_{\zeta\in
T^{1,0}M}{\mathfrak{R}}\left(\zeta,\bar{\zeta},\zeta,\bar{\zeta}\right),$

and if $h$ has non-positive Chern sectional curvature then

${\mathfrak{R}}\left(\xi,\bar{\xi},\eta,\bar{\eta}\right)\geq\mathop{{\rm
min\,}}\limits_{\zeta\in
T^{1,0}M}{\mathfrak{R}}\left(\zeta,\bar{\zeta},\zeta,\bar{\zeta}\right),$

where $\xi,\,\eta,\,\zeta$ are unit vectors. We complete the proof.∎

Acknowledge: This work is supported by the National Natural Science Foundation
of China (Grant No. 12001165).

## References

  * [1] Gray A. Curvature identities for Hermitian and almost Hermitian manifolds. Tohoku Math J 1976, 28(4), 601–612.
  * [2] Huybrechts D. Complex Geometry: An Introduction. Springer, Berlin Heidelber, 2005.
  * [3] Li H, Qiu C. A note on holomorphic sectional curvature of a Hermitian manifold. Glasgow Math J 2022, 64(3), 739–745.
  * [4] Liu K, Yang X. Geometry of Hermitian manifolds. Int J Math 2012, 23(6), 1250055(40 pages).
  * [5] Lu Q. A note on the extremum of sectional curvature of a Kähler manifold. Sci Sin Ser A 1984, 27(4), 367–371.
  * [6] Lu Q (Look K), Xu Y(Hsü I). A note on transitive domains. Acta Math Sin 1961, 11(1), 11–23; English transl., Chin J Math1962, 2, 11–26.
  * [7] Moroianu A. Lectures on Kähler geometry. Cambridge University Press, Cambridge, 2007.
  * [8] Siu Y. The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. Ann Math 1980, 112(1), 73–111(1980).
  * [9] Yang B, Zheng F. On curvature tensors of Hermitian manifolds. Commun. Anal Geom 2018, 26(5), 1195–1222.
  * [10] Zheng F. Complex differential geometry. American Mathematical Society, International Press, Boston, 2000.


# On biquandles for the groups $G_{n}^{k}$ and surface singular braid monoid

SANG YOUL LEE
Department of Mathematics, Pusan National University,
Busan 46241, Korea
<EMAIL_ADDRESS>
VASSILY OLEGOVICH MANTUROV
MIPT, Moscow,
Kazan Federal University,
North-Eastern University (China),
<EMAIL_ADDRESS>
and
IGOR MIKHAILOVICH NIKONOV
Department of Mechanics and Mathematics, Moscow State University,
Moscow, Russia.
<EMAIL_ADDRESS>
###### Abstract

The groups $G_{n}^{k}$ were defined by V. O. Manturov [16] in order to
describe dynamical systems in configuration systems. In the paper we consider
two applications of this theory: we define a biquandle structure on the groups
$G_{n}^{k}$, and construct a homomorphism from the surface singular braid
monoid to the group $G_{n}^{2}$.

##  1 Introduction

In 2015, V. O. Manturov [16] defined a series of groups $G_{n}^{k}$ for
natural numbers $n>k\geq 1$ and formulated the following principle:

If dynamical systems describing the motion of $n$ particles possess a nice
codimension one property depending exactly on $k$ particles, then these
dynamical systems admit a topological invariant valued in $G_{n}^{k}$.

These turn out to be good in studying fundamental groups of configuration and
moduli spaces when the initial space admits some geometry (for example, in the
Euclidean space one can study codimension 1 properties “three points are
collinear” or “four points belong to the same circle/line”). One of our goals
is to study when we have just topological spaces.

######  Definition 1.1.

The groups of free $k$-braids $G_{n}^{k}$ ($n>k\geq 1$) are defined as groups
with the set of generators $a_{m}$ which are indexed by all $k$-element
subsets of $\\{1,\ldots,n\\}$, and relations

  1. (1)

$(a_{m})^{2}=1$ for any unordered sets $m\subset\\{1,\ldots,n\\},Card(m)=k$;

  2. (2)

(far commutativity) $a_{m}a_{m^{\prime}}=a_{m^{\prime}}a_{m},$ if $Card(m\cap
m^{\prime})<k-1$;

  3. (3)

(tetrahedron relation) for every set
$U=\\{i_{1},\dots,i_{k+1}\\}\subset\\{1,\dots,n\\}$ of cardinality $(k+1)$,
let us denote them by $m^{p}=U\setminus\\{i_{p}\\},p=1,\dots,k+1$. Then

$a_{m^{1}}\cdots a_{m^{k+1}}=a_{m^{k+1}}\cdots a_{m^{1}}.$

We shall call the words in generators $a_{m}\in G_{n}^{k}$ diagrams of free
$k$-braids. These groups have become a sophisticated tool for researches in
knot theory and low dimensional topology [4].

######  Example 1.2.

Let $G^{2}_{n}$ be the group defined by the presentation generated by
$\\{a_{ij}\leavevmode\nobreak\ |\leavevmode\nobreak\
\\{i,j\\}\subset\\{1,\ldots,n\\},i<j\\}$ subject to the following relations:

  * (1)

$a^{2}_{ij}=1$ for all $i\not=j,$

  * (2)

$a_{ij}a_{kl}=a_{kl}a_{ij}$ for distinct $i,j,k,l,$

  * (3)

$a_{ij}a_{ik}a_{jk}=a_{jk}a_{ik}a_{ij}$ for distinct $i,j,k.$

In particular, the group $G^{2}_{3}$ is isomorphic to the group with a
presentation

$<a,b,c\leavevmode\nobreak\ |\leavevmode\nobreak\ a^{2}=b^{2}=(abc)^{2}=1>,$

where $a=a_{12},b=a_{13},c=a_{23}.$ Indeed, the relation $(abc)^{2}=1$ is
equivalent to the relation $abc=cab=1$ because of $a^{2}=b^{2}=c^{2}=1.$ This
obviously yields all the other tetrahedron relations.

######  Example 1.3.

Let $G^{3}_{n}$ be the group given by the presentation generated by
$\\{a_{ijk}\leavevmode\nobreak\ |\leavevmode\nobreak\
\\{i,j,k\\}\subset\\{1,\ldots,n\\},|\\{i,j,k\\}|=3\\}$ subject to the
following relations:

  * (1)

$a_{ijk}=1$ for all $\\{i,j,k\\}\subset\\{1,\ldots,n\\},$

  * (2)

$a_{ijk}a_{stu}=a_{stu}a_{ijk}$ if $\\{i,j,k\\}\cap\\{s,t,u\\}<2.$

  * (3)

$a_{ijk}a_{ijl}a_{ikl}a_{jkl}=a_{jkl}a_{ikl}a_{ijl}a_{ijk}$ for distinct
$i,j,k,l.$

Especially, the group $G^{3}_{4}$ is isomorphic to the group with a
presentation

$<a,b,c,d\leavevmode\nobreak\ |\leavevmode\nobreak\
a^{2}=b^{2}=c^{2}=1,(abcd)^{2}=1,(acdb)^{2}=1,(adbc)^{2}=1>.$

Here $a=a_{123},b=a_{124},c=a_{134},$ and $d=a_{234}.$

We can give a geometric interpretation to the free $k$-braids in the spirit of
Artin’s braids: Each free $k$-braid in $G_{n}^{k}$ given by a word
$w=a_{m_{1}}\dots
a_{m_{l}},m_{j}=\\{i_{j1},i_{j2},\dots,i_{jk}\\},j=1,\dots,l$ determines an
oriented graph $\Gamma_{w}$ with $n$ source vertices and $n$ sink vertices of
degree $1$, and $l$ vertices of degree $2k$ (with $k$ incoming and $k$
outcoming) edges, see Fig. 1. The vertices possesses the structure of opposite
edges: Each incoming edge is assigned to a unique outcoming edge (the opposite
edge).

Figure 1: A vertex corresponding to the generator $a_{ijk}$

The relations in the group $G_{n}^{k}$ induce transformations (moves) of the
free $k$-braid graphs, see Figs. 2–4 for the case $k=3$.

Figure 2: The involution relation $a_{ijk}^{2}=1$ Figure 3: The tetrahedron
relation $a_{ijk}a_{ijl}a_{ikl}a_{jkl}=a_{jkl}a_{ikl}a_{ijl}a_{ijk}$ Figure 4:
The far commutativity relation $a_{ijk}a_{klm}=a_{klm}a_{ijk}$

##  2 Biquandles for $G_{n}^{k}$Gnk

Biquandles were introduced in [5]. They can be described as rules for
colouring edges of link diagrams by elements of some set [3]. These rules
ensure that the sets of proper colourings of two equivalent link diagrams are
isomorphic. We can take this approach to give a definition of biquandles on
free $k$-braids.

######  Definition 2.1.

A biquandle for free $k$-braids (or a $k$-biquandle) is a pair $(X,B)$ where
$X$ is some set and $B\colon X^{\times k}\to X^{\times k}$ is a map that
satisfies the relations:

  1. 1.

$\pi\circ B=B\circ\pi$ for any permutation $\pi\in\Sigma_{k}$. Here we
consider the natural action of $\Sigma_{k}$ on $X^{\times k}$ by permutation
of the factors;

  2. 2.

$B^{2}=\mathop{id}$;

  3. 3.

$B_{\\{1,\dots,k\\}}B_{\\{p+1,\dots,k+p\\}}=B_{\\{p+1,\dots,k+p\\}}B_{\\{1,\dots,k\\}}$
for any $p=2,\dots,k-1$. Here

$B_{\\{i_{1},\dots,i_{k}\\}}\colon X^{\times 2k-1}\to X^{\times 2k-1}$

denotes the map which act as $B$ on the factors with indices from the set
$\\{i_{1},\dots,i_{k}\\}$ and as the identity on the others;

  4. 4.

$B_{\hat{1}}B_{\hat{2}}\cdots
B_{\widehat{k+1}}=B_{\widehat{k+1}}B_{\hat{k}}\cdots B_{\hat{1}}$ where
$B_{\hat{i}}=B_{\\{1,\dots,k+1\\}\setminus\\{i\\}}\colon X^{\times k+1}\to
X^{\times k+1}$.

The definition means that $B$ induces an action $\rho$ of the group
$G_{n}^{k}$ on the set $X^{\times n}$ with $\rho(a_{m})=B_{m}$.

######  Definition 2.2.

Let $(X,B)$ be a biquandle on free $k$-braids and $w$ be a diagram of free
$k$-braid (i.e. a word in generators $a_{m}$). Let $\Gamma_{w}$ be the
geometric realization of $w$, $E=E(\Gamma_{w})$ be its set of edges, and
$c\colon E\to X$ be a colouring of the edges with elements of $X$. We call the
colouring $c$ to be good if for each vertex of degree $2k$ the colours
$(x_{1},\dots,x_{k})$ of the incoming edges are related to the colours
$(y_{1},\dots,y_{k})$ of the opposite outcoming edges by the rule (see Fig. 5)

$(y_{1},\dots,y_{k})=B(x_{1},\dots,x_{k})\equiv(b_{1}(x_{1},\dots,x_{k}),\dots,b_{k}(x_{1},\dots,x_{k})).$
(2.1)

Figure 5: Colouring rule of the biquandle

We denote the set of good colourings by $Col_{(X,B)}(w)$. Given two $n$-tuples
of colours $\chi_{1},\chi_{2}\in X^{\times n}$, we define the coloring binding
number $col_{(X,B)}^{\chi_{1},\chi_{2}}(w)$ as the number of good colourings
of $w$ which have the colours $\chi_{1}$ on the incoming edges of the diagram
$\Gamma_{w}$ and the colours $\chi_{2}$ on the outcoming edges of
$\Gamma_{w}$.

Note that $col_{(X,B)}^{\chi_{1},\chi_{2}}(w)=0$ or $1$ because the colours of
the incoming edges determine uniquely the colours of the other edges of
$\Gamma_{w}$.

The following statement comes immediately from Definitions 1.1,2.1 and 2.2.

######  Theorem 2.3.

Let $w$ and $w^{\prime}$ be diagrams of free $k$-braids. If $w$ and
$w^{\prime}$ define the same element in $G_{n}^{k}$ then for any biquandle for
free $k$-braids $(X,B)$ there is a bijection between the sets of good
colourings of $w$ and of $w^{\prime}$ with elements of $X$.

Thus, the coloring binding numbers $col_{(X,B)}^{\chi_{1},\chi_{2}}$ are
invariants of free $k$-braids.

Now we give some examples of biquandles for free $k$-braids.

######  Example 2.4 (Fundamental $k$-biquandle).

Let $w$ be a diagram of free $k$-braid and $\Gamma_{w}$ be its geometric
realization. Analogously to the fundamental biquandle construction, we can
define the fundamental $k$-biquandle $BQ_{f}(w)=(X_{f}(w),B_{f}(w))$ for the
diagram $w$. The set $X_{f}(w)$ consists of equialence classes of formal words
which are results of all possible applications of a formal map $B_{f}(w)$ to
the elements of the set of edges $E(\Gamma_{w})$. The equivalence of the words
is induced by the relations of Definition 2.1 and the equation 2.1.

The fundamental $k$-biquandle can be characterized by the following statement.

######  Theorem 2.5.

  1. 1.

Let $w$ be a diagram of free $k$-braid. Then for any $k$-biquandle $(X,B)$
there is a natural bijection between the set of good colouring of $w$ with
elements of $X$ and the set of $k$-biquandle homomorphisms from the
fundamental $k$-biquandle $BQ_{f}(w)$ to the $k$-biquandle $(X,B)$.

  2. 2.

If two diagrams $w$ and $w^{\prime}$ of free $k$-braids define the same
element of $G_{n}^{k}$ then their fundamental $k$-biquandles $BQ_{f}(w)$ and
$BQ_{f}(w^{\prime})$ are isomorphic.

###### Proof.

The first statement of the theorem follows from the definition of the
fundamental biquandle. The second statement follows from the universal
property of the fundamental biquandle (first statement of the theorem) and
Theorem 2.3. $\square$

######  Example 2.6 (Gaussian $k$-biquandle).

Let $X={\mathbb{Z}}_{2}$ and
$B(x_{1},x_{2},\dots,x_{k})=(x_{1}+1,x_{2}+1,\dots,x_{k}+1)$. It is easy to
see $(X,B)$ satisfies the $k$-biquandle conditions.

The previous example can be generalized in the following way. Let $X$ be an
arbitrary set and $\tau\colon X\to X$ is an involution on $X$, i.e.
$\tau^{2}=\operatorname{id}_{X}$. Then the map

$B(x_{1},x_{2},\dots,x_{k})=(\tau(x_{1}),\tau(x_{2}),\dots,\tau(x_{k}))$

defines a $k$-biquandle structure on $X$ we call an involution $k$-biquandle.

We can give a further generalization.

Let $X$ be a set and $\tau\colon X\to X$ is an involution on $X$. Then $\tau$
is a product of independent transpositions $\tau=\prod_{i}(p_{i}q_{i})$. For
any tuple $\mathbf{x}=(x_{1},\dots,x_{k})$ we define its multiplicity vector
$\mathbf{m}(\mathbf{x})=(m_{1}(\mathbf{x}),m_{2}(\mathbf{x}),...)$ where
$m_{i}(\mathbf{x})$ is the number of elements $x_{j}$ in $\mathbf{x}$ which
are equal to $p_{i}$ or $q_{i}$ for some $i$. Let

$M_{k}=\\{(m_{1},m_{2},\dots)\,|\,m_{i}\in{\mathbb{Z}},m_{i}\geq 0,\
1\leq\sum_{i}m_{i}\leq k\\}$

be the set of possible notrivial multiplicity vectors and $\mu\subset M_{k}$
be an arbitrary subset of it. We define a map $B_{\tau,\mu}\colon X^{\times
k}\to X^{\times k}$ as following

$B_{\tau,\mu}(\mathbf{x})=\left\\{\begin{array}[]{cl}\tau(\mathbf{x}),&m(\mathbf{x})\in\mu,\\\
\mathbf{x},&m(\mathbf{x})\not\in\mu.\end{array}\right.$

Here $\tau(x_{1},\dots,x_{k})=(\tau(x_{1}),\dots,\tau(x_{k}))$.

######  Proposition 2.7.

The pair $(X,B_{\tau,\mu})$ is a $k$-biquandle.

###### Proof.

The first two properties of $k$-biquandle definition are evident. For the
tetrahedron and far commutativity relations note that if
$\mathbf{x}=(x_{1},\dots,x_{k})$ and $\mathbf{x}^{\prime}$ is obtained from
$\mathbf{x}$ by applying $\tau$ to some components of $\mathbf{x}$ then
$m(\mathbf{x})=m(\mathbf{x}^{\prime})$. This means the condition
$m(\mathbf{x})\in\mu$ does not change after applying operators $B_{\tau,\mu}$.
Therefore, for any subsets $m$ and $m^{\prime}$ with $k$ elements operators
$(B_{\tau,\mu})_{m}$ and $(B_{\tau,\mu})_{m^{\prime}}$ commute. Thus, the
tetrahedron and far commutativity relations hold. $\square$

We call the biquandle $(X,B_{\tau,\mu})$ a conditional involution
$k$-biquandle.

The direct computation shows the following.

######  Proposition 2.8.

Let $k=3$.

  1. 1.

Let $X={\mathbb{Z}}_{2}$. Then there is a unique up to isomorphism nontrivial
$3$-biquandle on $X$. This is the Gaussian biquandle.

  2. 2.

Let $X={\mathbb{Z}}_{3}$. Then there are 7 up to isomorphism nontrivial
$3$-biquandles on $X$. They are all conditional involution biquandles.

######  Example 2.9.

Let $X$ be a flat biquandle [11] with operations $(a,b)\mapsto(b\ast a,a\circ
b)$, see Fig. 6.

Figure 6: Flat biquandle

Then we can define a structure of $k$-biquandle $(X,B)$ on $X$ by the
splitting a $k$-crossing (see Fig. 7) into $\frac{k(k-1)}{2}$ double crossings
and applying the biquandle operations to the obtained diagram. The result does
not depend on the choice of the splitting.

Figure 7: Resolution of a $k$-crossing

For example, for $k=3$ we have the following formula

$B(x_{1},x_{2},x_{3})=((x_{1}\circ x_{2})\circ x_{3},(x_{2}\ast
x_{1})\circ(x_{3}\ast(x_{1}\circ x_{2})),(x_{3}\ast(x_{1}\circ
x_{2}))\ast(x_{2}\ast x_{1})).$

In order to the operator $B$ possesses the properties of Definition 2.1 we
need to impose the following relations on the flat biquandle operations:

  * •

$x=(x\circ y)\ast(y\ast x)$ for all $x,y\in X$;

  * •

$x\ast y=x\circ y$ for all $x,y\in X$

when $k\geq 2$, and additionally

  * •

$x\ast(y\circ z)=x\ast y$ for all $x,y,z\in X$;

  * •

$(x\circ y)\ast z=(x\circ z)\ast y$ for all $x,y,z\in X$

when $k\geq 3$.

##  3 Marked graph diagrams and surface-links

A marked graph is a graph $G$ possibly with $4$-valent vertices embedded in
$\mathbb{R}^{3}$ (or $S^{3}$) such that it has a finite number of vertices and
edges (possibly loops). Each vertex, say $v$, of $G$ is rigid, that is, there
exists a small rectangular neighborhood $N(v)$ homeomorphic to
$\\{(x,y)|-1\leq x,y\leq 1\\},$ where $v$ corresponds to the origin and the
edges incident to $v$ are represented by $x^{2}=y^{2}$, and also has a marker
represented by a thickened line segment on $N(v)$ with the core homeomorphic
to $\\{(x,0)|-\frac{1}{2}\leq x\leq\frac{1}{2}\\}$. A marked graph in
$\mathbb{R}^{3}$ can be presented as usual by a diagram on $\mathbb{R}^{2}$,
called a marked graph diagram (or ch-diagram), which is a classical link
diagram on $\mathbb{R}^{2}$ possibly with marked $4$-valent vertices look like
. Throughout this paper, a classical link (diagram) is regarded as a marked
graph (diagram) without marked vertices.

For a given marked graph diagram $D$, we define $L_{-}(D)$ and $L_{+}(D)$ to
be the link diagrams obtained from $D$ by replacing every marked vertex with
and , respectively, as illustrated in Figure 8. If $D$ has no marked vertices,
then we define $L_{-}(D)=L_{+}(D)=D$. We call $L_{-}(D)$ and $L_{+}(D)$ the
negative resolution and the positive resolution of $D$, respectively. A marked
graph diagram $D$ is said to be admissible if the negative resolution
$L_{-}(D)$ and the positive resolution $L_{+}(D)$ are all trivial link
diagrams.

Figure 8: Resolutions of marked graph diagrams

A surface-link (or knotted surface) is a closed 2-manifold smoothly (or
piecewise linearly and locally flatly) embedded in the standardly oriented
Euclidean $4$-dimensional space $\mathbb{R}^{4}$ (or the $4$-sphere $S^{4}$).
A connected surface-link is called a surface-knot. A $2$-sphere-link is
sometimes called a $2$-link. A connected $2$-link is also called a $2$-knot.
Two surface-links $\mathcal{L}$ and $\mathcal{L}^{\prime}$ in $\mathbb{R}^{4}$
are equivalent if they are ambient isotopic, i.e., there exists an orientation
preserving homeomorphism $h:\mathbb{R}^{4}\to\mathbb{R}^{4}$ such that
$h(\mathcal{L})=\mathcal{L}^{\prime}$. It is well known that every surface-
link is presented by an admissible marked graph diagram and vice versa (see
[10],[15],[18]).

The local transformations on marked graph diagrams depicted in Figure 9 are
called Yoshikawa moves. (These moves were first announced in [18].) The
Yoshikawa moves
$\Omega_{1},\Omega_{2},\Omega_{3},\Omega_{4},\Omega^{\prime}_{4}$ and
$\Omega_{5}$ are said to be of type I and the Yoshikawa moves
$\Omega_{6},\Omega^{\prime}_{6},\Omega_{7}$ and $\Omega_{8}$ are said to be of
type II. It is noted that all Yoshikawa moves of type I are realized to
ambient isotopies of $\mathbb{R}^{3}$, while all Yoshikawa moves of type II
are realized to ambient isotopies of $\mathbb{R}^{4}$, not $\mathbb{R}^{3}$
(see [10]).

Figure 9: Yoshikawa moves

Two marked graph diagrams $D$ and $D^{\prime}$ are said to be Yoshikawa move
equivalent if they can be transformed into each other by a finite sequence of
Yoshikawa moves of type I, type II and ambient isotopies of $\mathbb{R}^{2}$.

######  Theorem 3.1 ([13, 14, 17, 18]).

Let $\mathcal{L}$ and $\mathcal{L}^{\prime}$ be two surface-links and let $D$
and $D^{\prime}$ be two marked graph diagrams presenting $\mathcal{L}$ and
$\mathcal{L}^{\prime}$, respectively. Then $\mathcal{L}$ and
$\mathcal{L}^{\prime}$ are equivalent if and only if $D$ and $D^{\prime}$ are
Yoshikawa move equivalent.

##  4 Surface singular braid monoid $SSB_{n}$SSBn and groups $G^{k}_{n}$Gnk

A marked graph diagram is said to be in braid form if it is the geometric
closure of a singular braid on $n$ strands together with markers on its
singular vertices for some integer $n\geq 1$. (The notion of singular braid
was developed independently in [1] and [2].)

######  Proposition 4.1 ([7], Proposition 3.2).

For every surface-link, there exists its marked graph diagram in braid form.

Let $SSB_{n}(n\geq 1)$ denote the set of all marked graph diagrams in braid
form on $n$ strands. Then $SSB_{n}$ forms a monoid with some defining
relations (see Definition 4.2), which is called the surface singular braid
monoid on $n$ strands. The product $xy$ of two surface singular braids $x$ and
$y$ is obtained by putting them end to end as the product of two classical
braids. For $n=1$, this monoid is trivial with one element. Elements of
$SSB_{n}$ are called surface singular braids (on $n$ strands), and are
generated by four types of elementary surface singular braids
$a_{i},b_{i},c_{i},$ and $c_{i}^{-1}$ on $n$ strands for $i=1,\ldots,n-1$,
called the standard generators of $SSB_{n}$, where the correspondence of types
of crossings and types of marked singular vertices between $i$-th and
$(i+1)$-th strand are as shown in Fig. 10.

$\textstyle{\cdots}$$\textstyle{\cdots}$1ini+1$\textstyle{{}_{a_{i}}}$
$\textstyle{\cdots}$$\textstyle{\cdots}$1ini+1$\textstyle{{}_{b_{i}}}$

$\textstyle{\cdots}$$\textstyle{\cdots}$1ini+1$\textstyle{{}_{c_{i}}}$
$\textstyle{\cdots}$$\textstyle{\cdots}$1ini+1$\textstyle{{}_{c_{i}^{-1}}}$

Figure 10: The standard generators of $SSB_{n}$

######  Definition 4.2.

([7]) Let $n\in\mathbb{Z},n>1$, and $i,j,k\in\\{1,\ldots,n-1\\}$ such that
$|k-i|=1$, moreover let $x_{i},y_{i}\in\\{a_{i},b_{i},c_{i},c_{i}^{-1}\\}.$ We
define $SSB_{n}$ to be the monoid generated by $a_{i},b_{i},c_{i},$ and
$c_{i}^{-1}(1\leq i\leq n-1)$ subjected to the following defining relations:

  * (A1)

$c_{i}c_{i}^{-1}=1=c_{i}^{-1}c_{i}$,

  * (A2)

$x_{i}y_{j}=y_{j}x_{i}$ for $|i-j|>1$,

  * (A3)

$x_{i}c_{k}c_{i}=c_{k}c_{i}x_{k},$

  * (A4)

$x_{i}c_{k}^{-1}c_{i}^{-1}=c_{k}^{-1}c_{i}^{-1}x_{k},$

  * (A5)

$a_{i}b_{k}=b_{k}a_{i},$

  * (A6)

$a_{i}b_{i-2}(c_{i-1}c_{i-2}c_{i}c_{i-1})^{2}=a_{i}b_{i-2}$ for $i>2,$

  * (A7)

$b_{i}a_{i-2}(c_{i-1}c_{i-2}c_{i}c_{i-1})^{2}=b_{i}a_{i-2}$ for $i>2,$

  * (A8)

$a^{2}_{i}=a_{i},$

  * (A9)

$b^{2}_{i}=b_{i},$

  * (A10)

$a_{i}b_{i}c^{2}_{i}=a_{i}b_{i},$

  * (A11)

$a_{i}b_{k}(c_{i}c_{k}c_{i})^{2}=a_{i}b_{k}.$

For $x\in SSB_{n}$, we denote $\widehat{x}$ the closure of the surface
singular braid $x$ obtained by connecting the $n$ initial points to the
corresponding end-points by a collection of $n$ parallel strands as same as
the closure of classical braids. Note that the closed surface singular braid
$\widehat{x}$ is a marked graph diagram as discussed in the previous section
3. Let $CSB_{n}$ be the subset of $SSB_{n}$ consisting of only those elements
$x$ such that the closure $\widehat{x}$ is an admissible marked graph diagram,
i.e., both the positive resolution $L_{+}(\widehat{x})$ and the negative
resolution $L_{-}(\widehat{x})$ of $\widehat{x}$ are diagrams of trivial
classical links. We consider the following Markov type relations on surface
singular braids in $CSB_{n}$:

  * (C1)

$x_{i}S=Sx_{i}$ for $x_{i}\in\\{a_{i},b_{i},c_{i},c_{i}^{-1}\\}(1\leq i\leq
n-1)$ and $S\in CSB_{n},$

  * (C2)

$S=Sx_{n}$ for $S\in CSB_{n}$ and
$x_{n}\in\\{a_{n},b_{n},c_{n},c_{n}^{-1}\\}\subset CSB_{n+1}.$

######  Proposition 4.3.

([7]) Making change in a closed surface singular braid word formulation of a
surface-link by using one of the relations from (A1)-(A11) or (C1)-(C2), we
receive a formula of a surface-link of the same type.

We remark that Proposition 4.3 is a straightforward consequence of Theorem
3.1. The following question is still open.

Question 1. Whether any pair of marked graph diagrams in braid form of
equivalent surface-link can be transformed one another by using only relations
(A1)-(A11), (C1) and (C2)?

The following proposition gives another useful presentation for the surface
singular braid monoid $SSB_{n}$.

######  Proposition 4.4.

([8]) The monoid $SSB_{n}(n\geq 2)$ is generated by
$a_{i},b_{i},c_{i},c_{i}^{-1}(1\leq i\leq n-1)$ subjected to the following
relations:

  * (R1)

$c_{i}c^{-1}_{i}=1=c_{i}^{-1}c_{i},$

  * (R2)

$x_{i}y_{j}=y_{j}x_{i}$ for $|i-j|>1,$ where
$x_{i},y_{i}\in\\{a_{i},b_{i},c_{i},c_{i}^{-1}\\}$,

  * (R3)

$a_{i}c_{i}=c_{i}a_{i},$

  * (R4)

$b_{i}c_{i}=c_{i}b_{i},$

  * (R5)

$c_{i+1}c_{i}c_{i+1}=c_{i}c_{i+1}c_{i}$ for $1\leq i<n-1,$

  * (R6)

$a_{i+1}c_{i}c_{i+1}=c_{i}c_{i+1}a_{i}$ for $1\leq i<n-1,$

  * (R7)

$b_{i+1}c_{i}c_{i+1}=c_{i}c_{i+1}b_{i}$ for $1\leq i<n-1,$

  * (R8)

$a_{i}c_{i+1}c_{i}=c_{i+1}c_{i}a_{i+1}$ for $1\leq i<n-1,$

  * (R9)

$b_{i}c_{i+1}c_{i}=c_{i+1}c_{i}b_{i+1}$ for $1\leq i<n-1,$

  * (R10)

$a_{i}b_{i+1}=b_{i+1}a_{i}$ for $1\leq i<n-1,$

  * (R11)

$a_{i}b_{i}=b_{i}a_{i},$

  * (R12)

$a_{i}^{2}=a_{i},$

  * (R13)

$b_{i}^{2}=b_{i},$

  * (R14)

$a_{i}b_{i}c_{i}^{2}=a_{i}b_{i},$

  * (R15)

$a_{i}b_{i+1}(c_{i}c_{i+1}c_{i})^{2}=a_{i}b_{i+1}$ for $1\leq i<n-1,$

  * (R16)

$a_{i}b_{i+2}(c_{i+1}c_{i}c_{i+2}c_{i+1})^{2}=a_{i}b_{i+2}$ for $1\leq i<n-2.$

In the rest of the paper, we shall investigate some relationship between the
surface singular braid monoid $SSB_{n}$ and the groups $G_{m}^{k}$ of free
$k$-braids, and also $k$-biquandles for $G_{m}^{k}$ for some $m$ and $k$?. It
is natural to ask the following questions.

Question 2. Whether or not there is a homomorphism from $SSB_{n}$ to
$G_{m}^{k}$ for some $m$ and $k$?

We give a positive answer to this question in the next section.

##  5 Virtual surface singular braid monoid $VSSB_{n}$VSSBn

In this section we extend the surface singular braid monoid by adding a new
type of crossing — a virtual crossing (see Fig. 11). Then we define a
homomorphism from the constructed virtual surface singular braid monoid to the
group $G^{2}_{n}$ and describe the coloring invariants virtual surface
singular braids induced by this homomorphism.

Figure 11: Virtual crossing

######  Definition 5.1.

Let $n\geq 2$ be an integer. The virtual surface singular braid monoid
$VSSB_{n}$ is the monoid with the generators $a_{i},b_{i},c_{i},c_{i}^{-1}$
and $v_{i}$, $1\leq i\leq n-1$, and the relations (A1)-(A11) from Definition
4.2 and the virtual relations:

  * (V1)

$v_{i}x_{j}=x_{j}v_{i}$ for $|i-j|>1$ and
$x_{j}\in\\{a_{j},b_{j},c_{j},c_{j}^{-1},v_{j}\\}$,

  * (V2)

$v_{i}^{2}=1$,

  * (V3)

$x_{i}v_{i\pm 1}v_{i}=v_{i\pm 1}v_{i}x_{i\pm 1}$,
$x_{j}\in\\{a_{j},b_{j},c_{j},c_{j}^{-1},v_{j}\\},j=i,i\pm 1$

  * (V4)

$x_{i}v_{i}=v_{i}x_{i},$ $x_{i}\in\\{a_{i},b_{i}\\}$.

######  Remark 5.2.

Yoshikawa moves for virtual surface-links were found by L. Kauffman [12]. Here
we adapt those moves to the braid case. The relations (V2),(V3) correspond to
the detour move of Kauffman, and the relation (V4) is the commutation relation
between virtual crossings and marks.

Let $\Sigma_{n}$ be the group of permutations of the set $\\{1,\dots,n\\}$.
There is a homomorphism $\rho_{n}$ from the monoid $VSSB_{n}$ to the group
$\Sigma_{n}$:

$\rho_{n}(a_{i})=\rho_{n}(b_{i})=1,\quad\rho_{n}(c_{i})=\rho_{n}(c^{-1}_{i})=\rho_{n}(v_{i})=(i\
i+1),\quad i=1,\dots,n-1.$

Here $(i\ i+1)$ is the transposition of elements $i$ and $i+1$.

The kernel of the homomorphism $\rho_{n}$ is a submonoid $VPSSB_{n}$ in
$VSSB_{n}$. We call it the virtual pure surface singular braid monoid.

Now, let us define an action of the monoid $VSSB_{n}$ on the product
$G_{n}^{2}\times\Sigma_{n}$. Given an element $(g,\sigma)\in
G_{n}^{2}\times\Sigma_{n}$, we define

$\displaystyle
c_{i}\cdot(g,\sigma)=c_{i}^{-1}\cdot(g,\sigma)=(a_{\sigma(i),\sigma(i+1)}g,\sigma\cdot(i\
i+1)),$ $\displaystyle
a_{i}\cdot(g,\sigma)=b_{i}\cdot(g,\sigma)=(g,\sigma),\quad
v_{i}\cdot(g,\sigma)=(g,\sigma\cdot(i\ i+1)).$

Then for any $\beta\in VSSB_{n}$ we have

$\beta\cdot(1,1)=(\phi_{n}(\beta),\rho_{n}(\beta)).$ (5.2)

The direct computations show

######  Proposition 5.3.

1\. The map $\phi_{n}$ defines is a well-defines mapping of the monoid
$VSSB_{n}$ to the set $G_{n}^{2}$.

2\. The restriction of $\phi_{n}$ to the virtual pure surface singular braid
monoid $VPSSB_{n}$ is a homomorphism to the group $G_{n}^{2}$.

######  Corollary 5.4.

Let $(X,B)$ be a $2$-biquandle and $\chi_{1},\chi_{2}\in X^{\times n}$. Then
the colouring binding number
$col_{(X,B)}^{\chi_{1},\chi_{2}}(\phi_{n}(\beta))$, $\beta\in VSSB_{n}$, is an
invariant of virtual surface singular braids.

######  Example 5.5.

Consider the virtual surface singular braid
$\beta=c_{1}a_{2}v_{3}c_{2}^{-1}b_{1}c_{2}^{-1}$, see Fig. 12. Then
$\phi_{4}(\beta)=a_{12}a_{23}^{2}=a_{12}$. Take the Gaussian $2$-biquandle
$X={\mathbb{Z}}_{2}$, $B(x_{1},x_{2})=(x_{1}+1,x_{2}+1)$, and choose the
colours $4$-tuples $\chi_{1}=(0,1,0,1)$ and $\chi_{2}=(0,0,1,1)$. Then
$col_{(X,B)}^{\chi_{1},\chi_{2}}(\phi_{4}(\beta))=1$.

Figure 12: A virtual surface singular braid

Although we constructed a homomorphism from surface singular braids to
$G^{2}_{n}$, it is trivial in the sense that $\phi_{n}(\beta)=1$ if $\beta$ is
a classical braid, i.e. has no virtual crossings. Thus, the following question
remains actual.

Question 2’. How to construct a nontrivial homomorphism from $SSB_{n}$ to
$G_{m}^{k}$ for some $m$ and $k\geq 3$?

## Acknowledgements

The first author was supported under the framework of international
cooperation program managed by the National Research Foundation of Korea
(NRF-2019K2A9A1A06100201). The second and the third authors were supported by
the Russian Foundation for Basic Research (19-51-51004-NIF-a).

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