# PHG-Net: Persistent Homology Guided Medical Image Classification

Yaopeng Peng
University of Notre Dame
<EMAIL_ADDRESS>Hongxiao Wang
University of Notre Dame
<EMAIL_ADDRESS>Milan Sonka
University of Iowa
<EMAIL_ADDRESS>Danny Z. Chen
University of Notre Dame
<EMAIL_ADDRESS>This research was supported in part by NIH NIBIB Grant
R01-EB004640.

###### Abstract

Modern deep neural networks have achieved great successes in medical image
analysis. However, the features captured by convolutional neural networks
(CNNs) or Transformers tend to be optimized for pixel intensities and neglect
key anatomical structures such as connected components and loops. In this
paper, we propose a persistent homology guided approach (PHG-Net) that
explores topological features of objects for medical image classification. For
an input image, we first compute its cubical persistence diagram and extract
topological features into a vector representation using a small neural network
(called the PH module). The extracted topological features are then
incorporated into the feature map generated by CNN or Transformer for feature
fusion. The PH module is lightweight and capable of integrating topological
features into any CNN or Transformer architectures in an end-to-end fashion.
We evaluate our PHG-Net on three public datasets and demonstrate its
considerable improvements on the target classification tasks over state-of-
the-art methods.

##  1 Introduction

Deep neural networks (DNNs) are capable of learning useful image features
based on their potent representations, and are widely used in medical image
analysis. From AlexNet [17], VGG [25], to DenseNet [26, 10, 15], many
convolutional neural network (CNN) architectures have been proposed for rich
feature representations. Due to their nature, CNNs primarily focus on
capturing local features. Vision Transformers [7, 21] have been proposed to
procure global dependencies and long-range relationships by leveraging self-
attention mechanism, which allows models to attend to different patches and
learn their dependencies. However, these models tend to neglect key global and
robust anatomical structures (e.g., topological structures), such as connected
components, loops, and voids. Medical images commonly contain tissues, organs,
and lesions as connected components with specific patterns (e.g., loops and
voids), but such structures and typologies are often overlooked by deep
learning (DL) models.

In recent years, topological data analysis (TDA) [9] has been applied as a
powerful methodology to analyze data in chemistry [20], medicine [6], biology
[29], and other fields. Persistent homology (PH) is a most widely-used method
of TDA. It tracks topological changes of object dynamics during the filtration
process, where a lifespan is associated with these changes in the form of
entity birth or death. The collection of such birth-death time pairs forms a
persistence diagram (PD). But, effectively utilizing persistence diagrams in
machine learning is not straightforward due to the multi-set nature of the
persistent homology building process.

A technique was proposed to input topological signatures into DNNs for 2D
object shape and social network classification [12]. Additionally, a readout
operation to aggregate node features into a graph representation was
introduced [11]. Persistent landscapes were presented [2] as a method to
summarize topological data into vectors to use in machine learning. In [1],
persistent images were proposed as a stable representation that converts a
persistence diagram into a finite-dimensional vector representation. A
framework in [3] first encoded a persistence diagram into a vector and then
learned the vectorization using a neural network. Another method was developed
in [16] to first encode a persistence diagram into a list of persistent
landscapes and then vectorize them for use in DL models

Figure 1: Illustrating the pipeline of our proposed PHG-Net. The details of
the PD encoder are shown in Fig. 3(a). A fully connected block (FC block) is a
two-layer MLP shown in Fig. 3(b). The blue points in a persistence diagram
denote the 0-dimensional persistent homology (H0), and the orange points
denote the 1-dimensional persistent homology (H1). VisionLoss and TopoLoss
represent the vision loss and topological loss, respectively. $\odot$ denotes
matrix multiplication. We use a CNN backbone for illustration.

Medical images often exhibit intricate marked topological structures/patterns
of clinical targets, thereby rendering persistent homology (PH) a promising
computational technique that provides supplementary topological information
and insights alongside pixel-oriented CNNs [9]. More specifically, cubical
persistence leverages image intensity values as filtration functions in its
analytical process, which serves as a persistent homology tool that has been
proven to be highly useful for medical image analysis. For example, in [13],
it computed persistent curves and statistics, and these features were
integrated into ResNet for skin lesion classification. In [18], the
persistence of each region of interest (e.g., birth_time-death_time) was
ranked, and the images were clustered into sub-architectural groups. In [23],
topological features were combined with features extracted from the last CNN
layer for tumor segmentation in histology slides. In [27], it first masked out
the critical positions of breast MRI images using persistent homology and
classified the masked images. In [8], a persistence diagram was first encoded
into Betti curves and then integrated into a CNN network. In [28], it built
topological-attention of consecutive slices for 3D anisotropic image
segmentation. These methods often involved encoding a persistence diagram into
a vector using mathematical tools, which is usually a non-trivial process.
Another aspect that could be improved is that they concatenated topological
features with the last CNN layer, which incorporates another type of (CNN)
features, but lacks interactions between the topological features and multi-
scale CNN features. Furthermore, the topological features are not involved
when updating the gradients of the CNN or Transformer weights.

To address these issues, we propose a persistent homology-guided approach
(PHG-Net) for medical image classification. PHG-Net directly processes
persistence diagrams with a neural network and leverages the extracted
topological features in multiple layers of CNN or vision Transformer (not just
the last layer) to refine vision features at multiple scales. We treat
persistence diagrams as point clouds and process them with a network (called
the PH module or topological branch) inspired by PointNet [24]. Code is
available at https://github.com/yaoppeng/TopoClassification

Our main contributions are three-fold:

  1. 1.

Inspired by PointNet [24], we develop a new approach to process persistence
diagrams as point clouds rather than vectorized features, using a neural
network. The motivation for our work is that the differences among topological
structures of images can be represented by persistence diagrams and captured
by a neural network, thus guiding the learning of CNNs or Transformers. This
process is data-driven and learnable.

  2. 2.

Our PHG-Net is capable of incorporating topological features into any base
vision models, including CNNs and Transformers.

  3. 3.

Our new PHG-Net approach achieves considerable improvements on three public
datasets compared to state-of-the-art medical image classification methods.

##  2 Method

We use a CNN backbone to illustrate our PHG-Net. The overall pipeline of our
PHG-Net approach is shown in Fig. 1. Given an image $I$, the persistence
diagram (PD) of $I$ is first computed, and a PointNet-like neural network is
then applied to encode the PD into a feature vector. The feature vector is
integrated into feature maps generated by the CNN at each of its ConvBlocks
for feature map refinement.

In this section, we first provide a brief review of cubical persistent
homology (the readers may refer to [9] for further exposition). We then
describe how we design a neural network encoder to directly convert a PD into
a feature vector. Finally, we present the process of using persistent homology
to guide the CNN for feature refinement in medical image classification.

Figure 2: Illustrating the process of sub-level filtration. (a) A 3D plot of
the sub-level set filtration of pixel intensities of a crop in an H&E stained
image. The corresponding hematoxylin channel of the crop is projected on the
$xy$ plane. The $z$-axis is for threshold values. The red plane corresponds to
one 2D slice of the 3D plot at threshold = 150. (b) Three thresholded binary
images of the cropped image. As the threshold value increases or decreases,
some connected components or loops are born or die. (c) The computed
persistence diagram of the image crop (blue points denote the 0-D persistent
homology (H0), and orange points denote the 1-D persistent homology (H1)).

###  2.1 Cubical Persistent Homology

Persistent homology (PH) is a powerful mathematical tool for analyzing
topological properties of data. It is capable of identifying and quantifying
important features of data that persist over a range of different spatial
scales, thus providing insight into the underlying structures of the data.
Fig. 2 illustrates the filtration process of PH. Given an image $I$ and a
series of threshold values $(\tau_{0},\tau_{1},\ldots,\tau_{h})$,
$\tau_{0}<\tau_{1}<\cdots<\tau_{h}$, in each step $i$, we threshold $I$ as a
sub-level image: $S_{\tau_{i}}=\\{x:x\in I,f(x)\leq\tau_{i}\\}$, where $f$:
$I\rightarrow R$ is a function of pixel intensities. That is, for each
$\tau_{i}$, a sub-image $S_{\tau_{i}}$ is generated, and its connected
components, loops, and voids are recorded. The sequence of $\\{S_{\tau_{i}}\
|\ i=0,1,\ldots,h\\}$, with $S_{\tau_{0}}\subseteq
S_{\tau_{1}}\subseteq\cdots\subseteq S_{\tau_{h}}$, forms the filtration.
Following the evolution of these sub-level images through the threshold
sequence, the homology groups are induced as
$\\{H(S_{\tau_{0}}),H(S_{\tau_{1}}),\ldots,H(S_{\tau_{h}})\\}$, where
$H(S_{\tau_{i}})$ records the topological features of $S_{\tau_{i}}$ (e.g.,
connected component, loops, voids). When a new topological structure appears
or is “born” at threshold $\tau_{i}$ and disappears, “dies”, or merges with
another topological structure at threshold $\tau_{j}$, a tuple
$(\tau_{i},\tau_{j})$ is recorded and plotted as a 2D point with $\tau_{i}$ on
the $x$-axis and $\tau_{j}$ on the $y$-axis; $\tau_{j}-\tau_{i}$ denotes the
lifespan (i.e., persistence) of that topological structure. A structure with a
long persistence means that the differences between it and its surroundings
are significant and tend to be more salient (e.g., the boundary region of an
image).

###  2.2 Persistence Diagram Encoder

Since PD is a most commonly used descriptor for persistent homology, there are
an increasing number of methods to map PDs into a vector representation for
machine learning tasks. However, these methods often rely on a mathematical
encoding step, which is usually non-trivial and specifically designed, and may
result in loss of certain information of the PDs. To address this issue, we
propose a data-driven method to encode PDs.

Specifically, we treat each data point in a PD as a 2D feature, denoted as
$(f_{born},f_{death})$. A straightforward method would be to aggregate all the
data points and obtain a 2D vector, for example, through max pooling or
average pooling. However, such an operation may lead to losing too much
information of the PD and may not effectively represent the entire filtration
process. Therefore, we first increase the dimension of each data point so that
the points can interact with one another. Then, we aggregate the information
of all the points into a vector representation.

One key property of the points in a PD is permutation invariance, which means
that the feature vector of the PD remains the same if some of the points
exchange positions with one another. Note that this property is different from
that of processing image data with a CNN, where exchanging pixels may result
in different outputs. Based on this observation, we formulate the PD-encoding
problem as:

$f(p_{1},p_{2},\ldots,p_{n})=g(m(p_{1}),m(p_{2}),\ldots,m(p_{n})),$ (1)

where $p_{i}$ denotes a data point in a PD, $f(p_{0},p_{1},\ldots,p_{n-1})$
represents the output feature vector, and $m(p_{i})$ is a function that maps a
data point into a higher dimension so that it contains interactions of the
data points and the information of the entire PD. To account for the
permutation invariance property, we use a multi-layer perceptron (MLP) instead
of using convolution. In Eq. (1), $g$ is a function that aggregates
information from all the points, such as through max pooling or average
pooling. The entire vector learning process is carried out with a neural
network, which is data-driven, and the vectorization process is supervised by
the target task. Fig. 3(a) illustrates the details of the persistence diagram
encoder that we design.

Figure 3: Illustrating (a) our persistence diagram encoder and (b) a
persistent homology guided residual block (FC block). $\odot$ and $\oplus$ are
matrix multiplication and summation, respectively.

For persistent homology, let $H_{0}$ denote the connected components, $H_{1}$
denote the loops, and $H_{2}$ denote the voids in the 3D representation.
However, a neural network taking points from these groups as input cannot
differentiate which homology group that a point belongs to. Therefore, we add
an additional one-hot vector to label each point. In other words, a point
$(birth,death)$ in $H_{i}$ is represented as
$(birth,death,0,\ldots,0,1,0,\ldots,0)$, in which the only 1 is located in the
$i$-th position of the 0-1 sequence.

###  2.3 Persistent Homology Guidance (PHG)

The persistent homology features capture global topological information of an
image. Previous methods typically utilized these features by integrating them
with features learned by CNNs or Transformers. For example, in [13], it
concatenated PH features and CNN features just before the classification step.
In [8], PH features were treated as a teacher for adjusting CNN features.
However, in concatenation based methods, gradient values of multiple branches
during back-propagation are computed separately, and there are no direct
interactions between the gradients of different branches. Moreover,
concatenation just before the classification step lacks interactions between
topological features and CNN features at multi-scale levels. To tackle this
issue, we propose integrating topological supervision with each CNN or
Transformer block. Additionally, we add a fully connected block (FC block) to
regulate the intervention of the topological branch in each CNN or Transformer
block (see Fig. 3(b)). This also enables us to refine the feature maps in
multi-scales using topological features.

Given an intermediate CNN feature map $F$, $F\in\mathbb{R}^{H\times W\times
C}$, our PHG-Net integrates a topological feature vector $t$ into $F$, where
$t\in\mathbb{R}^{M}$ is the output of the persistent homology encoder (see Eq.
(1)). To ensure flexibility and applicability to any feature map, we employ a
gate mechanism in the process:

$t^{\prime}=f(t,W)=\sigma(W_{1}(Relu(W_{2}(t)))),$ (2)

where $t^{\prime}\in R^{C}$ is the processed topological feature vector that
will be used to refine the feature map $F$, $t$ denotes the output of the PD
encoder, $W_{1}\in R^{\frac{C}{r}\times C}$ and $W_{2}\in
R^{M\times\frac{C}{r}}$ are the parameters of two MLP layers, $\sigma$ is the
sigmoid activation function, and $r$ is a parameter for balancing the trade-
off between the performance and complexity. Afterwards, the topological
guidance is formulated as:

$F^{\prime}=F\otimes t^{\prime},$ (3)

where $F^{\prime}$ is the refined feature map and $\otimes$ is element-wise
multiplication. During the refinement, the feature vector $t^{\prime}\in
R^{C}$ will first be broadcast along the spatial dimension of $F^{\prime}$.
Fig. 3(b) illustrates a persistent homology guided residual block.

###  2.4 Loss Function

The whole model is optimized with the following loss ${\cal{L}}$:

${\cal{L}}={\cal{L}}_{V}(\tilde{y}_{v},y)+\alpha{\cal{L}}_{Topo}(\tilde{y}_{topo},y),$
(4)

where ${\cal{L}}_{V}$ and ${\cal{L}}_{Topo}$ are the cross-entropy losses of
the vision model and topological branches respectively; $\tilde{y}_{v}$,
$\tilde{y}_{topo}$, and $y$ denote the vision branch output, topological
branch output, and ground truth respectively, and $\alpha$ is a hyper-
parameter for balancing the CNN and topological losses. See Fig. 1 for more
details.

##  3 Experiments

###  3.1 Datasets

We evaluate our proposed PHG-Net approach using the following three datasets.

ISIC 2018: A skin lesion dataset for predicting 7 classes of skin disease
lesions. It consists of 10,015 training images and 193 validation images [5];
the test set is unavailable. For fair comparison, we follow the dataset split
strategy in [30], with which five-fold cross validation is conducted.

Prostate Cancer Classification: A dataset of hematoxylin and eosin (H&E)
stained prostate cancer images that consists of 77 whole slide images (WSIs)
from 19 patients [18]. The WSIs are divided into regions of interest (RoIs) of
$512\times 512$ pixels each. 5,182 RoIs are generated and three classes of
prostate cancer are to be predicted. We also conduct five-fold cross
validation.

CBIS-DDSM: A Curated Breast Imaging Subset of Digital Database for Screening
Mammography (CBIS-DDSM). It contains 1,566 participants and 6,775 studies,
which are categorized as benign or malignant [19]. We use the train/test split
given by the dataset provider, in which 20% of the cases are for testing and
the rest for training.

###  3.2 Experimental Setup

Table 1: Experimental results on the ISIC 2018 dataset. Reported $p$-values reflect statistical significance of the improvement achieved by SwinV2-B + PHG over SwinV2-B, assessed by paired t-test; “–” is reported if SwinV2-B + PHG performance is lower than SwinV2-B for a specific evaluation metric.  Method | Accuracy | AUC | Sensitivity | Specificity
---|---|---|---|---
ResNet152 [10] | 88.83$\pm$0.45 | 97.52$\pm$0.15 | 80.85$\pm$0.16 | 97.01$\pm$0.28
ResNet152 [10] \+ PHG | 90.03$\pm$0.24 | 98.14$\pm$0.23 | 83.75$\pm$0.30 | 97.12$\pm$0.33
SENet154 [14] | 89.80$\pm$0.37 | 97.92$\pm$0.27 | 81.96$\pm$0.37 | 96.90$\pm$0.25
SENet154 [14] \+ PHG | 90.83$\pm$0.23 | 98.67$\pm$0.30 | 82.04$\pm$0.41 | 97.12$\pm$0.19
SwinV2-B [21] | 90.85$\pm$0.34 | 97.99$\pm$0.38 | 82.23$\pm$0.27 | 97.32$\pm$0.27
SwinV2-B [21] \+ PHG | 91.92$\pm$0.27 | 98.97$\pm$0.42 | 83.14$\pm$0.36 | 97.28$\pm$0.35
$p$-value | 0.0006 | 0.005 | 0.002 | –
Table 2: Experimental results on the Prostate Cancer dataset. Reported $p$-values reflect statistical significance of the improvement achieved by SwinV2-B + PHG over SwinV2-B, assessed by paired t-test.  Method | Accuracy | AUC | Sensitivity | Specificity
---|---|---|---|---
ResNet152 [10] | 92.96$\pm$0.28 | 98.02$\pm$0.21 | 96.54$\pm$0.18 | 97.44$\pm$0.34
ResNet152 [10] \+ PHG | 94.01$\pm$0.28 | 98.80$\pm$0.30 | 96.91$\pm$0.22 | 96.67$\pm$0.41
SENet154 [14] | 93.73$\pm$0.39 | 98.67$\pm$0.34 | 94.84$\pm$0.50 | 96.38$\pm$0.28
SENet154 [14] \+ PHG | 97.99$\pm$0.21 | 99.72$\pm$0.29 | 96.55$\pm$0.33 | 98.26$\pm$0.40
SwinV2-B [21] | 95.21$\pm$0.33 | 98.61$\pm$0.31 | 97.22$\pm$0.34 | 97.99$\pm$0.29
SwinV2-B [21] \+ PHG | 98.64$\pm$0.27 | 99.83$\pm$0.24 | 98.34$\pm$0.29 | 98.17$\pm$0.17
$p$-value | $\ll 0.001$ | 0.0001 | 0.0005 | 0.265
Table 3: Experimental results on the CBIS-DDSM dataset demonstrate that SwinV2-B + PHG significantly outperforms all the other tested methods for all the evaluation metrics. The $p$-values are obtained by paired t-test between SwinV2-B + PHG and SwinV2-B. Method | Accuracy | AUC | Sensitivity | Specificity
---|---|---|---|---
ResNet152 [10] | 71.16$\pm$0.34 | 78.75$\pm$0.45 | 73.19$\pm$0.55 | 71.65$\pm$1.08
ResNet152 [10] \+ PHG | 74.34$\pm$0.31 | 79.35$\pm$0.36 | 71.74$\pm$1.10 | 72.03$\pm$0.49
SENet154 [14] | 71.96$\pm$0.19 | 79.01$\pm$0.29 | 72.02$\pm$0.34 | 71.52$\pm$0.62
SENet154 [14] \+ PHG | 75.66$\pm$0.26 | 82.23$\pm$0.42 | 73.53$\pm$0.29 | 73.07$\pm$0.21
SwinV2-B [21] | 73.51$\pm$0.22 | 81.58$\pm$0.54 | 72.92$\pm$0.31 | 72.28$\pm$0.20
SwinV2-B [21] \+ PHG | 77.23$\pm$0.37 | 83.39$\pm$0.43 | 75.89$\pm$0.29 | 74.83$\pm$0.38
$p$-value | $\ll 0.001$ | 0.0004 | $\ll 0.001$ | $\ll 0.001$
Table 4: Ablation study on the ISIC 2018 dataset. Reported $p$-$\text{value}_{1}$ was obtained using paired t-test between settings 12 and 13 in this table, measuring the statistical significance of the differences between KD and our PHG. “–” is reported since the Spe score of 13 is lower than 12. Reported $p$-$\text{value}_{2}$ was obtained using paired t-test between settings 10 and 13, measuring the statistical significance between PLLay and our PD encoder. “Concat” means concatenating CNN features and topological features at the last CNN layer.  | Backbone | PD Module | PD Fusion | Acc | AUC | Sen | Spe
---|---|---|---|---|---|---|---
| SENet154 | SwinV2-B | PersLay | PLLay | Our PD | Concat | KD | PHG
1 |  |  | $\checkmark$ |  |  |  |  |  | 65.69$\pm$0.26 | 82.30$\pm$0.55 | 36.48$\pm$0.21 | 91.72$\pm$0.36
2 |  |  |  | $\checkmark$ |  |  |  |  | 67.13$\pm$0.26 | 82.02$\pm$0.49 | 32.12$\pm$0.25 | 91.97$\pm$0.48
3 |  |  |  |  | $\checkmark$ |  |  |  | 70.72$\pm$0.43 | 91.66$\pm$0.52 | 53.01$\pm$0.72 | 94.62$\pm$0.36
4 | $\checkmark$ |  | $\checkmark$ |  |  |  |  | $\checkmark$ | 90.02$\pm$0.68 | 97.37$\pm$0.38 | 81.80$\pm$0.61 | 97.01$\pm$0.70
5 | $\checkmark$ |  |  | $\checkmark$ |  |  |  | $\checkmark$ | 90.27$\pm$0.21 | 97.04$\pm$0.65 | 80.66$\pm$0.71 | 96.82$\pm$0.45
6 | $\checkmark$ |  |  |  | $\checkmark$ | $\checkmark$ |  |  | 89.97$\pm$0.34 | 97.34$\pm$0.47 | 82.21$\pm$0.57 | 96.50$\pm$0.39
7 | $\checkmark$ |  |  |  | $\checkmark$ |  | $\checkmark$ |  | 90.27$\pm$0.29 | 96.94$\pm$0.48 | 82.09$\pm$0.37 | 96.47$\pm$0.64
8 | $\checkmark$ |  |  |  | $\checkmark$ |  |  | $\checkmark$ | 90.83$\pm$0.23 | 98.67$\pm$0.30 | 82.04$\pm$0.41 | 97.12$\pm$0.19
9 |  | $\checkmark$ | $\checkmark$ |  |  |  |  | $\checkmark$ | 91.00$\pm$0.44 | 98.23$\pm$0.17 | 81.76$\pm$0.19 | 97.40$\pm$0.25
10 |  | $\checkmark$ |  | $\checkmark$ |  |  |  | $\checkmark$ | 91.12$\pm$0.29 | 98.46$\pm$0.31 | 82.33$\pm$0.52 | 96.78$\pm$0.41
11 |  | $\checkmark$ |  |  | $\checkmark$ | $\checkmark$ |  |  | 91.18$\pm$0.19 | 98.09$\pm$0.26 | 81.99$\pm$0.24 | 97.68$\pm$0.37
12 |  | $\checkmark$ |  |  | $\checkmark$ |  | $\checkmark$ |  | 91.00$\pm$0.44 | 98.20$\pm$0.38 | 82.42$\pm$0.59 | 97.45$\pm$0.40
13 |  | $\checkmark$ |  |  | $\checkmark$ |  |  | $\checkmark$ | 91.92$\pm$0.27 | 98.97$\pm$0.42 | 83.14$\pm$0.36 | 97.28$\pm$0.35
$p$-$\text{value}_{1}$ | 0.016 | 0.004 | 0.048 | –
$p$-$\text{value}_{2}$ | 0.002 | 0.06 | 0.021 | 0.072
Table 5: Ablation study on the Prostate Cancer dataset. Reported $p$-$\text{value}_{1}$ was obtained using paired t-test between settings 12 and 13 in this table, measuring the statistical significance between KD and our PHG. Reported $p$-$\text{value}_{2}$ was obtained using paired t-test between settings 10 and 13, measuring the statistical significance between PLLay and our PD encoder. “Concat” means concatenating CNN features and topological features at the last CNN layer. | Backbone | PD Module | PD Fusion | Acc | AUC | Sen | Spe
---|---|---|---|---|---|---|---
| SENet154 | SwinV2-B | PersLay | PLLay | Our PD | Concat | KD | PHG
1 |  |  | $\checkmark$ |  |  |  |  |  | 75.31$\pm$0.46 | 86.43$\pm$0.53 | 67.37$\pm$0.57 | 84.42$\pm$0.38
2 |  |  |  | $\checkmark$ |  |  |  |  | 77.34$\pm$0.30 | 89.84$\pm$0.56 | 66.58$\pm$0.33 | 85.75$\pm$0.22
3 |  |  |  |  | $\checkmark$ |  |  |  | 82.35$\pm$0.46 | 92.53$\pm$0.20 | 76.37$\pm$0.48 | 87.46$\pm$0.58
4 | $\checkmark$ |  | $\checkmark$ |  |  |  |  | $\checkmark$ | 96.91$\pm$0.43 | 99.68$\pm$0.25 | 95.92$\pm$0.40 | 98.09$\pm$0.44
5 | $\checkmark$ |  |  | $\checkmark$ |  |  |  | $\checkmark$ | 97.11$\pm$0.30 | 99.68$\pm$0.43 | 96.13$\pm$0.47 | 98.21$\pm$0.39
6 | $\checkmark$ |  |  |  | $\checkmark$ | $\checkmark$ |  |  | 97.02$\pm$0.55 | 99.45$\pm$0.39 | 96.28$\pm$0.30 | 98.40$\pm$0.47
7 | $\checkmark$ |  |  |  | $\checkmark$ |  | $\checkmark$ |  | 96.57$\pm$0.29 | 99.55$\pm$0.27 | 96.57$\pm$0.29 | 97.91$\pm$0.28
8 | $\checkmark$ |  |  |  | $\checkmark$ |  |  | $\checkmark$ | 97.99$\pm$0.21 | 99.72$\pm$0.29 | 96.55$\pm$0.33 | 98.26$\pm$0.40
9 |  | $\checkmark$ | $\checkmark$ |  |  |  |  | $\checkmark$ | 98.21$\pm$0.31 | 99.54$\pm$0.46 | 97.54$\pm$0.38 | 97.59$\pm$0.24
10 |  | $\checkmark$ |  | $\checkmark$ |  |  |  | $\checkmark$ | 98.04$\pm$0.26 | 99.06$\pm$0.27 | 97.31$\pm$0.24 | 97.54$\pm$0.19
11 |  | $\checkmark$ |  |  | $\checkmark$ | $\checkmark$ |  |  | 98.04$\pm$0.29 | 98.12$\pm$0.25 | 98.45$\pm$0.41 | 97.12$\pm$0.28
12 |  | $\checkmark$ |  |  | $\checkmark$ |  | $\checkmark$ |  | 98.11$\pm$0.40 | 99.75$\pm$0.21 | 97.24$\pm$0.32 | 97.77$\pm$0.22
13 |  | $\checkmark$ |  |  | $\checkmark$ |  |  | $\checkmark$ | 98.64$\pm$0.27 | 99.83$\pm$0.24 | 98.34$\pm$0.29 | 98.17$\pm$0.17
$p$-$\text{value}_{1}$ | 0.039 | 0.59 | 0.0004 | 0.012
$p$-$\text{value}_{2}$ | 0.007 | 0.001 | 0.0002 |  $\ll$0.001
Table 6: Ablation study on the CBIS-DDSM dataset. Reported $p$-$\text{value}_{1}$ was obtained using paired t-test between settings 12 and 13 in this table, measuring the statistical significance between KD and our PHG. Reported $p$-$\text{value}_{2}$ was obtained using paired t-test between settings 10 and 13, measuring the statistical significance between PLLay and our PD encoder. “Concat” means concatenating CNN features and topological features at the last CNN layer. | Backbone | PD Module | PD Fusion | Acc | AUC | Sen | Spe
---|---|---|---|---|---|---|---
| SENet154 | SwinV2-B | PersLay | PLLay | Our PD | Concat | KD | PHG
1 |  |  | $\checkmark$ |  |  |  |  |  | 61.38$\pm$0.32 | 55.05$\pm$0.10 | 55.60$\pm$0.25 | 57.72$\pm$0.14
2 |  |  |  | $\checkmark$ |  |  |  |  | 63.23$\pm$0.26 | 63.06$\pm$0.21 | 60.36$\pm$0.37 | 59.40$\pm$0.35
3 |  |  |  |  | $\checkmark$ |  |  |  | 64.02$\pm$0.31 | 62.44$\pm$0.48 | 58.29$\pm$0.44 | 60.17$\pm$0.19
4 | $\checkmark$ |  | $\checkmark$ |  |  |  |  | $\checkmark$ | 74.87$\pm$0.30 | 79.71$\pm$0.33 | 74.00$\pm$0.35 | 73.13$\pm$1.31
5 | $\checkmark$ |  |  | $\checkmark$ |  |  |  | $\checkmark$ | 74.60$\pm$0.40 | 81.66$\pm$0.32 | 75.02$\pm$0.25 | 73.69$\pm$0.35
6 | $\checkmark$ |  |  |  | $\checkmark$ | $\checkmark$ |  |  | 72.36$\pm$0.64 | 80.27$\pm$0.57 | 73.58$\pm$0.47 | 72.89$\pm$0.46
7 | $\checkmark$ |  |  |  | $\checkmark$ |  | $\checkmark$ |  | 73.01$\pm$0.47 | 81.90$\pm$0.35 | 74.40$\pm$0.46 | 73.01$\pm$0.77
8 | $\checkmark$ |  |  |  | $\checkmark$ |  |  | $\checkmark$ | 75.66$\pm$0.26 | 82.23$\pm$0.42 | 73.53$\pm$0.29 | 73.07$\pm$0.21
9 |  | $\checkmark$ | $\checkmark$ |  |  |  |  | $\checkmark$ | 73.71$\pm$0.30 | 81.26$\pm$0.27 | 72.64$\pm$0.29 | 72.54$\pm$0.30
10 |  | $\checkmark$ |  | $\checkmark$ |  |  |  | $\checkmark$ | 74.06$\pm$0.45 | 82.21$\pm$0.53 | 73.45$\pm$0.47 | 73.04$\pm$0.42
11 |  | $\checkmark$ |  |  | $\checkmark$ | $\checkmark$ |  |  | 73.58$\pm$0.27 | 82.04$\pm$0.51 | 74.11$\pm$0.45 | 73.44$\pm$0.51
12 |  | $\checkmark$ |  |  | $\checkmark$ |  | $\checkmark$ |  | 74.34$\pm$0.40 | 82.41$\pm$0.43 | 74.03$\pm$0.46 | 73.87$\pm$0.29
13 |  | $\checkmark$ |  |  | $\checkmark$ |  |  | $\checkmark$ | 77.23$\pm$0.37 | 83.39$\pm$0.43 | 75.89$\pm$0.29 | 74.83$\pm$0.38
$p$-$\text{value}_{1}$ |  $\ll$0.001 | 0.007 |  $\ll$0.001 | 0.002
$p$-$\text{value}_{1}$ |  $\ll$0.001 | 0.005 |  $\ll$0.001 | 0.0001

Common data augmentation techniques such as random flipping, color jittering,
random rotation, and cropping are used. The network is optimized using the
Adam optimizer with an initial learning rate of 0.0001, $\beta_{1}$ value of
0.9, and $\beta_{2}$ value of 0.999. A polynomial learning rate decay with a
power of 0.9 is applied. The maximum number of training epochs is set to 1000.
The value of $\alpha$ in Eq. (4) is set to 0.1. The parameter $r$ for
balancing the trade-off between performance and complexity is empirically set
to 8. All experiments are conducted on an NVIDIA P100 GPU using PyTorch. Four
common evaluation metrics are used: accuracy (Acc), area under the receiver
operating characteristic (ROC) curve (AUC), sensitivity (Sen), and specificity
(Spe).

We utilize the GUDHI package [22] to compute persistence diagrams. The points
in a persistence diagram are first scaled to the range of $[0,1]$ and then
normalized. Additionally, a one-hot marker indicating which homology group
each point belongs to is added. Any persistence value smaller than 10 is
ignored. The points within each homology group are sorted in decreasing order
based on their persistence values. For each persistence diagram, we
experimented with selecting the top $n$ = 50, 100, 150, 200, and 250 points
(see Eq. (1)). We found that selecting $n$ = 150 points yields saturated
performance while maintaining relatively low computation, as most of the
points are very close to the diagonal line and are considered as noise points.

Moreover, we add dummy points of $(0,0)$ to a persistence diagram if the
number of points in it is less than 150. All the points are given an
additional coordinate to indicate that a point is an added one (marked by 0)
or originally presented one (marked by 1).

Table 7: Time and parameter complexities of our PHG-Net approach on an NVIDIA P100 GPU; w/ share and w/o share represent using one PD encoder and multiple PD encoders for multi-scale vision feature fusion, respectively. Method | Input Size | Acc | # Params. | FLOPs | FPS
---|---|---|---|---|---
ISIC 2018 | Prostate | CBIS-DDSM
SwinV2-B | (3, 224, 224) | 90.85 | 95.21 | 73.51 | 86.913 M | 20.370 G | 14.937
SwinV2-B+PLLay+PHG | (3, 224, 224) (4, 300) | 91.12 | 98.04 | 74.06 | 95.073 M | 20.375 G | 20.787
SwinV2-B+PD+PHG w/ share | (3, 224, 224) (4, 300) | 91.92 | 98.64 | 77.23 | 98.065 M | 20.796 G | 15.914
SwinV2-B+PD+PHG w/o share | (3, 224, 224) (4, 300) | 91.95 | 98.70 | 77.04 | 108.892 M | 22.062 G | 17.059

###  3.3 Results and Analysis

We evaluate the effectiveness and robustness of our PHG-Net approach by
utilizing two common CNN backbones (ResNet152 [10] and SENet154 [14]), which
performed well on medical image datasets (e.g., used by the team attaining
top-1 score in the ISIC 2018 Challenge), as well as a latest Transformer, Swin
Transformer v2 (SwinV2-B) [21]. These models are evaluated without using extra
training data. To demonstrate the statistical significance of the performance
improvements yielded by our PHG-Net, we compute $p$-values (with paired
$t$-test) between SwinV2-B + PHG (our model) and SwinV2-B by running each
experiment 5 rounds. For each round, we select 5 epochs that achieve the top-5
best accuracy and take their average as the final result for robust
evaluation.

The ISIC 2018 and Prostate Cancer datasets are for multi-class classification
tasks. Thus, we report the OvR (One-vs-Rest) scores for AUC, Specificity, and
Sensitivity. In each time, we take one class as positive and the other classes
as negative, under which we compute the score for each class $c$. Finally, the
average score of all the classes is reported.

Tables 1, 2, and 3 present the experimental results. We find that our PHG-Net
approach improves, for example, the accuracy by 1.07%, 3.43%, and 3.72% on the
ISIC 2018, prostate cancer, and CBIS-DDSM datasets, respectively, when using
the SwinV2-B backbone. These performance improvements and the $p$-values
verify the effectiveness of our approach, especially on the prostate cancer
and breast mammography datasets, where discernible changes in appearances or
topological structures occur when cancer is present. For instance, in an
H&E-stained prostate image, the number of nuclei is likely to increase if
cancer cells are found in the image. In breast mammography images, tumor areas
are often characterized by dense masses or clusters of micro-calcifications.
In some cases, they may be present as tiny calcium deposits that appear as
small white specks or clusters in the images. These topological structures,
such as connected components or loops, can differentiate the images containing
them from normal images.

Figure 4: Validation losses of different persistent homology based methods and
the KD (with our PD encoder) method.

###  3.4 Ablation Study

Figure 5: Qualitative examples of different persistent homology based methods
and the KD (with our PD encoder) method. The 1st, 2nd, and 3rd rows are for
samples from the ISIC 2018, Prostate Cancer, and CBIS-DDSM datasets,
respectively. Columns (a)-(e) are for input images, heat-maps by PLLay,
PersLay, KD, and PHG-Net, respectively. Green and red correspond to correct
and incorrect predictions, respectively. The green or red floating-point
number under each heat-map represents the top-1 prediction confidence (the
higher the better).

To examine the effects of our new persistence diagram (PD) encoder and PHG
mechanism, we compare our methods with two common persistence diagram
vectorization-based methods: (1) PersLay [3] (it encodes a persistence diagram
and then refines the PD using an FC layer, where the persistence silhouette
setting [4] is applied), and (2) PLLay [16] (it utilizes a weighted persistent
landscape to extract topological features). Specifically, we pass the
topological features extracted by these methods through the PH branch and
compare their performances. Furthermore, we investigate the effect of using
only topological features in the models. Additionally, we compare our approach
with a typical knowledge distillation method (KD) [8], but replace its
backbone with SENet154 and SwinV2-B, respectively, for fair comparison.

Tables 4, 5, and 6 show that our PD encoder outperforms the PersLay and PLLay
methods, while our PHG outperforms the KD and concatenation (Concate, which
concatenates CNN features and topological features at the last CNN layer)
methods, since our PHG refines CNN features at multiple scales. This confirms
the effectiveness of our schemes for constructing the PD encoder and PH
guidance. It is worth noting that the methods relying solely on topological
information (PersLay, PLLay, and our PD encoder) do not perform well on these
datasets. Note that PersLay is a general form of persistence landscape [2],
persistence silhouette [4], and persistence images [1] by utilizing different
hyper-parameters, and we found that there is no obvious difference among the
settings of these PH methods. Thus, our approach also outperforms these PH
methods. The $p$-values in Tables 4, 5, and 6 demonstrate the statistical
significance of our PD encoder and PHG module.

###  3.5 Time and Parameter Complexity

To examine the time and parameter complexities of our PHG-Net, we present the
number of parameters, floating point operations (FLOPs), and frames per second
(FPS) for our approach and the SwinV2-B model in Table 7 for comparison.

In Table 7, $(3,224,224)$ denotes the size of an input image, while $(4,300)$
stands for the size of the corresponding persistence diagram. It can be
observed from Table 7 that the additional parameters and computation costs
caused by our PHG-Net are quite limited. Furthermore, sharing the PD Encoder
at multiple scales will result in fewer parameters and FLOPs without reducing
the performance.

###  3.6 Qualitative Results

The validation losses of different persistent homology based methods and the
KD method on the three datasets using the backbone of SwinV2-B [14] are shown
in Fig. 4.

Some qualitative examples based on the backbone of SwinV2-B on the three
datasets are shown in Fig. 5, which demonstrate that our PHG-Net is capable of
helping the network concentrate on the correct regions.

##  4 Conclusions

In this paper, we proposed to extract and encode persistence diagrams of
medical images as important topological features for classification tasks
using a neural network (PD encoder) that we designed. Our new approach, PHG-
Net, is data-driven, learnable, and superior in performance over known
topology based methods. We further developed a persistence diagram (PD) guided
mechanism, which incorporates PD features into CNN or Transformer for co-
training of DL models. Experiments on three datasets validated the
effectiveness of our PHG-Net. Complexity analysis showed that the extra costs
introduced by our approach are quite small. It is expected that our PHG-Net
approach will provide a new perspective and paradigm for combining topological
features with CNNs/Transformers to improve AI model performances for medical
image analysis tasks.

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# Verbal quandles with one parameter

Elizaveta Markhinina  Timur Nasybullov

###### Abstract

We find all words $W(x,y,z)$ in the free group $F(x,y,z)$, such that for every
group $G$ and an element $c\in G$ the algebraic system $(G,*_{W,c})$ with the
binary operation $*_{W,c}$ given by $a*_{W,c}b=W(a,b,c)$ for $a,b\in G$ is a
quandle. Such quandles are called verbal quandles with one parameter.

_Keywords:_ quandle, verbal quandle, group.

##  1 Introduction and preliminaries

A quandle is an algebraic system with a binary operation whose three axioms
encode the Reidemeister moves on knot diagrams. More formally, a quandle $Q$
is an algebraic system with one binary algebraic operation $(x,y)\mapsto x*y$
which satisfies the following three axioms:

  * (q1)

$x*x=x$ for all $x\in Q$,

  * (q2)

the map $S_{x}:y\mapsto y*x$ is a bijection of $Q$ for all $x\in Q$,

  * (q3)

$(x*{y})*z=(x*z)*({y*z})$ for all $x,y,z\in Q$.

First examples of quandles date back to 1940s when Takasaki defined keys,
objects which were later known as involutory quandles [22]. The concept of
quandles was explicitly presented in the independent works of Joyce [20] and
Matveev [21]. To each oriented diagram $D_{K}$ of an oriented knot $K$ in
$\mathbb{R}^{3}$ they associate the quandle $Q(K)$ which does not change if we
apply the Reidemeister moves to the diagram $D_{K}$. Joyce and Matveev proved
that two knot quandles $Q(K_{1})$ and $Q(K_{2})$ are isomorphic if and only if
there is a (probably, reversing orientation) homeomorphism of the ambient
space $\mathbb{R}^{3}$ which maps $K_{1}$ to $K_{2}$. The knot quandle is a
very strong invariant for knots in $\mathbb{R}^{3}$, however, usually it is
very difficult to determine if two knot quandles are isomorphic. In [12] it is
shown that the isomorphism problem for quandles is, from the perspective of
Borel reducibility, fundamentally difficult (Borel complete). Sometimes
homomorphisms from knot quandles to simpler quandles provide useful
information that helps determine whether two knot quandles are isomorphic.
This potential utility leads to the necessity of constructing quandles which
are convenient to work with.

Another area where quandles find applications is the theory of the set-
theoretical Yang-Baxter equation. The Yang-Baxter equation first appeared in
theoretical physics and statistical mechanics in the works of Yang [23] and
Baxter [13, 14]. The notion of the set-theoretical Yang-Baxter equation, was
introduced by V. Drinfel’d in the context of quantum groups (see [17]). Recall
that a set-theoretic solution of the Yang-Baxter equation is a pair $(X,r)$,
where $X$ is a set and $r:X\times X\to X\times X$ is a bijective map such that

$(r\times id)(id\times r)(r\times id)=(id\times r)(r\times id)(id\times r).$

If $Q$ is a quandle with the operation $*$, then the pair $(Q,r)$, where
$r:Q\times Q\to Q\times Q$ is the map given by

$r(x,y)=(y*x,x)$ (1)

for $x,y\in Q$ is a set-theoretic solution of the Yang-Baxter equation.

In the recent papers [6, 8] there was constructed a general method how a given
solution of the set-theoretical Yang-Baxter equation on an arbitrary algebraic
system $X$ can be used for constructing a representation of the virtual braid
group $VB_{n}$ by automorphisms of the algebraic system $X$. In the papers [2,
4] these this method was used for constructing new representations of vertual
braid groups. In the papers [7, 8] there was introduced a method how a given
solution of the set-theoretical Yang-Baxter equation on an algebraic system
$X$ can be used for constructing an invariant for virtual knots and links
which is an algebraic system from the same category as $X$. If $X$ is a
quandle, and the solution of the set-theoretical Yang-Baxter equation on $X$
is given by (1), then using procedures described in [6, 7, 8] it is possible
to construct a representation $VB_{n}\to{\rm Aut}(X)$, and a quandle invariant
for virtual links. However, in order to apply procedures described in [6, 7,
8] to $X$ and $r$, the quandle $X$ must be cinvenient to work with, hence it
is necessary to construct such quandles.

Quandles were also investigated from an algebraic point of view and relations
to other algebraic structures such as Lie algebras [16], Frobenius algebras
and Yang-Baxter equation [16], Hopf algebras [1], quasigroups and Moufang
loops [18], representation theory [19] and ring theory [11]. In all of these
areas it is always useful to have examples of quandles which are convenient to
work with.

A lot of examples of quandles which are convenient to work with come from
groups. The most common example of a quandle is the conjugation quandle ${\rm
Conj}(G)$ of a group $G$, i. e. the quandle $(G,*)$ with $x*y=y^{-1}xy$ for
$x,y\in G$. If we define another operation $*$ on the group $G$, namely
$x*y=yx^{-1}y$, then the set $G$ with this operation also forms a quandle.
This quandle is called the core quandle of a group $G$ and is denoted by ${\rm
Core}(G)$. In particular, if $G$ is an abelian group, then the quandle ${\rm
Core}(G)$ is called the Takasaki quandle of the abelian group $G$ and is
denoted by $T(G)$. Such quandles were studied by Takasaki in [22]. Other
examples of quandles arising from groups can be found, for example, in [3, 5,
9].

The so-called verbal quandls were introduced in the paper [10]. Let $w=w(x,y)$
be an element of the free group $F(x,y)$ with two generators. If $G$ is a
group, then $w$ defines a map $w:G\times G\to G$ which maps a pair $(a,b)$ to
$w(a,b)$. For $a,b\in G$ denote by $a*_{w}b=w(a,b)$. We can think about
$*_{w}$ as about new binary operation on the group $G$, so, $(G,*_{w})$ is an
algebraic system. If the algebraic system $(G,*_{w})$ is a quandle, then we
call this quandle a verbal quandle defined by the word $w$. It is clear that
${\rm Conj}(G)$ and ${\rm Core}(G)$ are verbal quandles. The following theorem
which classifies verbal quandles is proved in [10].

######  Theorem 1.

Let $w\in F_{2}$ be an element from the free group. If $Q=(G,*_{w})$ is a
quandle for every group $G$, then one of the following conditions holds

  1. 1.

$w(x,y)=yx^{-1}y$,

  2. 2.

$w(x,y)=y^{-s}xy^{s}$ for $s\in\mathbb{Z}$.

In the present paper we introduce the notion of a verbal quandle with $n$
parameters which generalizes the notion of a verbal quandle. Let
$W=W(x,y,z_{1},z_{2},\dots,z_{n})$ be a word from the free group with $(n+2)$
free generators. For a group $G$ and fixed elements
$c_{1},c_{2},\dots,c_{n}\in G$ denote by $*_{W,c_{1},c_{2},\dots,c_{n}}$ the
binary operation on $G$ defined by the rule

$a*_{W,c_{1},c_{2},\dots,c_{n}}b=W(a,b,c_{1},c_{2},\dots,c_{n}).$

If $(G,*_{W,c_{1},c_{2},\dots,c_{n}})$ is a quandle, then this quandles is
called a verbal quandle with $n$ parameters. If $n=0$, then this quandle is
just a verbal quandle. The main result of the present paper is the following
theorem which classifies verbal quandles with one parameter.

######  Theorem 2.

Let $W=W(x,y,z)$ be an element from the free group $F(x,y,z)$. If
$Q=(G,*_{W,c})$ is a quandle for every group $G$ and an element $c\in G$, then
one of the following conditions holds

  1. 1.

$W(x,y,z)=yx^{-1}y$,

  2. 2.

$W(x,y,z)=y^{-s}xy^{s}$ for $s\in\mathbb{Z}$,

  3. 3.

$W(x,y,z)=yz^{-s}y^{-1}xz^{s}$ for $s\in\mathbb{Z}$,

  4. 4.

$W(x,y,z)=yz^{-s}xy^{-1}z^{s}$ for $s\in\mathbb{Z}$,

  5. 5.

$W(x,y,z)=z^{-s}y^{-1}xz^{s}y$ for $s\in\mathbb{Z}$,

  6. 6.

$W(x,y,z)=z^{-s}xy^{-1}z^{s}y$ for $s\in\mathbb{Z}$.

It is clear that Theorem 1 follows from Theorem 2. In the consequent paper we
are going to apply solutions of the set-theoretic Yang-Baxter equation
constructed by verbal quandles with one parameter using formula (1) in order
to construct new group invariants for virtual knots and links using
constructions from [6, 7, 8].

Acknowledgment. The work is supported by Mathematical Center in Akademgorodok
under agreement 075-15-2022-281 with the Ministry of Science and Higher
Education of the Russian Federation.

##  2 Proof of the main theorem

Let
$W=x^{\alpha_{1}}y^{\beta_{1}}z^{\gamma_{1}}x^{\alpha_{2}}y^{\beta_{2}}z^{\gamma_{2}}\ldots
x^{\alpha_{k}}y^{\beta_{k}}z^{\gamma_{k}}$ for
$\alpha_{i},\beta_{i},\gamma_{i}\in\mathbb{Z}$ be a reduced word in the free
group $F(x,y,z)$. Since $(G,*_{W,c})$ is a quandle, from the axiom (q2) of a
quandle follows that for every $a,b\in G$ there exists an element $d\in G$
such that

$d*_{W,c}a=W(d,a,c)=d^{\alpha_{1}}a^{\beta_{1}}c^{\gamma_{1}}d^{\alpha_{2}}a^{\beta_{2}}c^{\gamma_{2}}\ldots
d^{\alpha_{k}}a^{\beta_{k}}c^{\gamma_{k}}=b.$ (2)

It meant that we can express $d$ from equation (2) via $a,b,c$. It is possible
if and only if $W=u(y,z)x^{\varepsilon}w(y,z)$ for $u(y,z),w(y,z)\in F(y,z)$,
$\varepsilon\in\\{\pm 1\\}$. From the axiom (q1) it follows that for every
element $a\in G$ the equality $a*_{W,c}a=a$ holds. This equality can be
rewritten in details in the following form

$a=a*_{W,c}a=W(a,a,c)=u(a,c)a^{\varepsilon}w(a,c),$

and therefore $u(a,c)=a(w(a,c))^{-1}a^{-\varepsilon}$ for every group $G$,
fixed element $c\in G$ and element $a\in G$. Hence we have
$u(y,z)=y(w(y,z))^{-1}y^{-\varepsilon}$ and

$W(x,y,z)=y(w(y,z))^{-1}y^{-\varepsilon}x^{\varepsilon}w(y,z).$ (3)

Let us understand what are the possible options for the word $w(y,z)\in
F(y,z)$. In order to do this express the word $w(y,z)$ as a reduced word

$w=y^{q_{1}}z^{s_{1}}y^{q_{2}}z^{s_{1}}\ldots y^{q_{n}}y^{s_{n}},$

where $q_{i},s_{i}\in\mathbb{Z}$ are non-zero integers with possible
exceptions for $q_{1}$ and $s_{n}$. Denote by $|w(y,z)|$ the syllabic length
of the word $w(y,z)$. Depending on the word $w(y,z)$ and the parameter
$\varepsilon\in\\{\pm 1\\}$ we consider the cases collected in the following
table.

$|w|$ | $w\in F(y,z)$ | $\varepsilon\in\\{\pm 1\\}$ | $q_{i},s_{i}\in\mathbb{Z}$
---|---|---|---
1: $|w|=0$ | $1$ |  |
2: $|w|=1$ | 2.1: $y^{q}$ |  |
| 2.2: $z^{s}$ | 2.2.1: $\varepsilon=1$ |
|  | 2.2.2: $\varepsilon=-1$ |
3: $|w|=2$ | 3.1: $y^{q}z^{s}$ | 3.1.1: $\varepsilon=1$ | 3.1.1.1: $q=-1$
|  |  | 3.1.1.2: $q\neq-1$
|  | 3.1.2: $\varepsilon=-1$ | 3.1.2.1: $q=1$
|  |  | 3.1.2.2: $q\neq 1$
| 3.2: $z^{s}y^{q}$ | 3.2.1: $\varepsilon=1$ | 3.2.1.1: $q=1$
|  |  | 3.2.1.2: $q\neq 1$
|  | 3.2.2: $\varepsilon=-1$ |
4: $|w|=3$ | 4.1: $y^{q_{1}}z^{s}y^{q_{2}}$ | 4.1.1: $\varepsilon=1$ | 4.1.1.1: $q_{2}=1$, $q_{1}=-1$
|  |  | 4.1.1.2: $q_{2}=1$, $q_{1}\neq-1$
|  |  | 4.1.1.3: $q_{2}\neq 1$
|  | 4.1.2: $\varepsilon=-1$ | 4.1.2.1: $q_{2}=1$
|  |  | 4.1.2.2: $q_{2}\neq 1$
| 4.2: $z^{s_{1}}y^{q}z^{s_{2}}$ |  | 4.2.1: $|q|\neq 1$
|  |  | 4.2.2: $|q|=1$
5: $|w|\geq 4$ |  |  |

From this table it is clear that the cases we are going to consider are all
possible cases (the empty fields mean that the corresponding parameters can be
arbitrary). We are going to prove that for the word $W(x,y,z)$ given by
formula (3) the algebraic system $Q=(G,*_{W,c})$ is a quandle for every group
$G$ and an element $c\in G$ if and only if the word $w(y,z)$ in formula (3) is
given by one of the cases 1, 2.1, 2.2.1, 3.1.1.1, 3.2.1.1, 4.1.1.1. These
cases give six possibilities for the word $W(x,y,z)$ described in the
formulation of the theorem.

Case 1: $|w(y,z)|=0$. In this case $w(y,z)=1$, and the word

$W(x,y,z)=y^{1-\varepsilon}x^{\varepsilon}$

doesn’t depend on $z$, hence for every group $G$ we have
$(G,*_{W,c})=(G,*_{W})$. From Theorem 1 it follows that we have either
$W(x,y,z)=yx^{-1}y$, or $W(x,y,z)=y^{-s}xy^{s}$ for $s\in\mathbb{Z}$. It means
that in Case 1 we have a quandle.

Case 2: $|w(y,z)|=1$. In this case we have either $w(y,z)=y^{q}$ for non-zero
$q\in\mathbb{Z}$, or $w(y,z)=z^{s}$ for non-zero $s\in\mathbb{Z}$.

If $w(y,z)=y^{q}$ for non-zero $q\in\mathbb{Z}$ (case 2.1), then the word

$W(x,y,z)=y^{1-q-\varepsilon}x^{\varepsilon}y^{q}$

doesn’t depend on $z$, hence for every group $G$ we have
$(G,*_{W,c})=(G,*_{W})$. From Theorem 1 it follows that we have either
$W(x,y,z)=yx^{-1}y$, or $W(x,y,z)=y^{-s}xy^{s}$ for $s\in\mathbb{Z}$. It means
that in Case 2.1 we have a quandle.

If $w(y,z)=z^{s}$ for non-zero $s\in\mathbb{Z}$ (case 2.2), then
$W(x,y,z)=yz^{-s}y^{-\varepsilon}x^{\varepsilon}z^{s}$. Let us understand when
the operation $*_{W,c}$ satisfies the third quandle axiom (q3) for each group
$G$ and an element $c\in G$

$\displaystyle(a*_{W,c}b)*_{W,c}d$ $\displaystyle=(W(a,b,c))*_{W,c}d$
$\displaystyle=(bc^{-s}b^{-\varepsilon}a^{\varepsilon}c^{s})*_{W,c}d$
$\displaystyle=W(bc^{-s}b^{-\varepsilon}a^{\varepsilon}c^{s},d,c)$
$\displaystyle=dc^{-s}d^{-\varepsilon}(bc^{-s}b^{-\varepsilon}a^{\varepsilon}c^{s})^{\varepsilon}c^{s},$
(4) $\displaystyle(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$
$\displaystyle=(W(a,d,c))*_{W,c}(W(b,d,c))$
$\displaystyle=(dc^{-s}d^{-\varepsilon}a^{\varepsilon}c^{s})*_{W,c}(dc^{-s}d^{-\varepsilon}b^{\varepsilon}c^{s})$
$\displaystyle=W(dc^{-s}d^{-\varepsilon}a^{\varepsilon}c^{s},dc^{-s}d^{-\varepsilon}b^{\varepsilon}c^{s},c)$
$\displaystyle=(dc^{-s}d^{-\varepsilon}b^{\varepsilon}c^{s})c^{-s}(dc^{-s}d^{-\varepsilon}b^{\varepsilon}c^{s})^{-\varepsilon}(dc^{-s}d^{-\varepsilon}a^{\varepsilon}c^{s})^{\varepsilon}c^{s}.$
(5)

If $\varepsilon=1$ (case 2.2.1), then from formulas (4), (5) it follows that

$\displaystyle(a*_{W,c}b)*_{W,c}d$
$\displaystyle=dc^{-s}d^{-1}bc^{-s}b^{-1}ac^{2s},$
$\displaystyle(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$
$\displaystyle=dc^{-s}d^{-1}bc^{s}c^{-s}c^{-s}b^{-1}dc^{s}d^{-1}dc^{-s}d^{-1}ac^{s}c^{s}$
$\displaystyle=dc^{-s}d^{-1}bc^{-s}b^{-1}ac^{2s},$

i. e. the equality $(a*_{W,c}b)*_{W,c}d=(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$ holds
independently on $s$. It means that in Case 2.2.1 we have a quandle. If
$\varepsilon=-1$ (case 2.2.2), then from formulas (4), (5) it follows that

$\displaystyle(a*_{W,c}b)*_{W,c}d$
$\displaystyle=dc^{-s}dc^{-s}ab^{-1}c^{s}b^{-1}c^{s},$ (6)
$\displaystyle(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$
$\displaystyle=dc^{-s}db^{-1}c^{s}c^{-s}dc^{-s}db^{-1}c^{s}c^{-s}ad^{-1}c^{s}d^{-1}c^{s}$
$\displaystyle=dc^{-s}db^{-1}dc^{-s}db^{-1}ad^{-1}c^{s}d^{-1}c^{s}.$ (7)

Let us look to $a,b,c,d$ as to the generators of the free group $F(a,b,c,d)$
on four generators. In this situation from equalities (6), (7) it is clear
that

$(a*_{W,c}b)*_{W,c}d\neq(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$

independently on $s$, i. e. there exists a group $G=F(a,b,c,d)$ and an element
$c\in G$ such that in $(G,*_{W,c})$ the third quandle axiom (q3) doen’t hold,
i. e. $(G,*_{W,c})$ is not a quandle. It means that in Case 2.2.2 we don’t
have a quandle.

Case 3: $|w(y,z)|=2$. In this case $w(y,z)$ is either equal to
$w(y,z)=y^{q}z^{s}$ or to $w(y,z)=z^{s}y^{q}$ for non-zero integers
$q,s\in\mathbb{Z}$.

If $w(y,z)=y^{q}z^{s}$ (case 3.1), then
$W(x,y,z)=yz^{-s}y^{-q-\varepsilon}x^{\varepsilon}y^{q}z^{s}$. Let us
understand when the operation $*_{W,c}$ satisfies the third quandle axiom (q3)
for each group $G$ and an element $c\in G$

$\displaystyle(a*_{W,c}b)*_{W,c}d$ $\displaystyle=(W(a,b,c))*_{W,c}d$
$\displaystyle=(bc^{-s}b^{-q-\varepsilon}a^{\varepsilon}b^{q}c^{s})*_{W,c}d$
$\displaystyle=W(bc^{-s}b^{-q-\varepsilon}a^{\varepsilon}b^{q}c^{s},d,c)$
$\displaystyle=dc^{-s}d^{-q-\varepsilon}(bc^{-s}b^{-q-\varepsilon}a^{\varepsilon}b^{q}c^{s})^{\varepsilon}d^{q}c^{s},$
(8) $\displaystyle(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$
$\displaystyle=(W(a,d,c))*_{W,c}(W(b,d,c))$
$\displaystyle=(dc^{-s}d^{-q-\varepsilon}a^{\varepsilon}d^{q}c^{s})*_{W,c}(dc^{-s}d^{-q-\varepsilon}b^{\varepsilon}d^{q}c^{s})$
$\displaystyle=W(dc^{-s}d^{-q-\varepsilon}a^{\varepsilon}d^{q}c^{s},dc^{-s}d^{-q-\varepsilon}b^{\varepsilon}d^{q}c^{s},c)$
$\displaystyle=(dc^{-s}d^{-q-\varepsilon}b^{\varepsilon}d^{q}c^{s})c^{-s}(dc^{-s}d^{-q-\varepsilon}b^{\varepsilon}d^{q}c^{s})^{-q-\varepsilon}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\cdot(dc^{-s}d^{-q-\varepsilon}a^{\varepsilon}d^{q}c^{s})^{\varepsilon}(dc^{-s}d^{-q-\varepsilon}b^{\varepsilon}d^{q}c^{s})^{q}c^{s}.$
(9)

Let us consider the case when $\varepsilon=1$ (case 3.1.1). From equalities
(8), (9) we see that the equality
$(a*_{W,c}b)*_{W,c}d=(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$ holds if an only if

$c^{-s}b^{-q-1}ab^{q}c^{s}d^{q}=d^{q}(dc^{-s}d^{-q-1}bd^{q}c^{s})^{-q-1}(dc^{-s}d^{-q-1}ad^{q}c^{s})(dc^{-s}d^{-q-1}bd^{q}c^{s})^{q}.$
(10)

If $q=-1$ (case 3.1.1.1), then equality (10) clearly holds, i. e. for
$W(x,y,z)=yz^{-s}xy^{-1}z^{s}$ the algebraic system $(G,*_{W,c})$ is a quandle
for every group $G$ and an element $c\in G$. It means that in Case 3.1.1.1 we
have a quandle. If $q\neq 1$ (case 3.1.1.2), i. e. $q<-1$ or $q>0$, then look
to $a,b,c,d$ as to the generators of the free group $F(a,b,c,d)$. The reduced
word on the left side of equality (10) begins with $c^{-s}$, while the word on
the right side of equality (10) begins either with $d^{q+1}$ (if $q<-1$) or
with $d^{q}$ (of $q>0$), hence equality (10) doesn’t hold. It means that in
Case 3.1.1.2 we don’t have a quandle.

Let us consider the case when $\varepsilon=-1$ (case 3.1.2). From equalities
(8), (9) we see that the equality
$(a*_{W,c}b)*_{W,c}d=(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$ holds if an only if

$c^{-s}b^{-q}ab^{q-1}c^{s}b^{-1}d^{q}=\\\
=b^{-1}d^{q}(dc^{-s}d^{-q+1}b^{-1}d^{q}c^{s})^{-q+1}c^{-s}d^{-q}ad^{q-1}c^{s}d^{-1}(dc^{-s}d^{-q+1}b^{-1}d^{q}c^{s})^{q}.$
(11)

If $q=1$ (case 3.1.2.1), then equality (11) can be rewritten in the following
form

$c^{-s}b^{-1}ac^{s}b^{-1}d=b^{-1}dc^{-s}d^{-1}ab^{-1}dc^{s}.$

Looking to $a,b,c,d$ as to the generators of the free group it is clear that
the last equality doesn’t hold for all non-zero integers $s$, hence, if $q=1$,
then there exists a group $G$ and an element $c\in G$ such that $(G,*_{W,c})$
is not a quandle. It means that in Case 3.1.2.1 we don’t have a quandle. If
$q\neq 1$, i. e. $q<0$ or $q>1$ then look to $a,b,c,d$ as to the generators of
the free group. The reduced word on the left side of equality (11) begins with
$c^{-s}$, while the word on the right side of equality (11) begins with
$b^{-1}$, hence, equality (11) doesn’t hold. So, in this case there exists a
group $G$ and an element $c\in G$ such that $(G,*_{W,c})$ is not a quandle. It
means that in Case 3.1.2.2 we don’t have a quandle.

Let us now consider the case when $|w(y,z)|=2$ and $w(y,z)=z^{s}y^{q}$ for
non-zero integers $q,s\in\mathbb{Z}$ (case 3.2). In this case the word
$W(x,y,z)$ has the following form

$W(x,y,z)=y^{1-q}z^{-s}y^{-\varepsilon}x^{\varepsilon}z^{s}y^{q}.$

Let us understand when the operation $*_{W,c}$ satisfies the third quandle
axiom (q3) for each group $G$ and an element $c\in G$

$\displaystyle(a*_{W,c}b)*_{W,c}d$ $\displaystyle=(W(a,b,c))*_{W,c}d$
$\displaystyle=(b^{1-q}c^{-s}b^{-\varepsilon}a^{\varepsilon}c^{s}b^{q})*_{W,c}d$
$\displaystyle=W(b^{1-q}c^{-s}b^{-\varepsilon}a^{\varepsilon}c^{s}b^{q},d,c)$
$\displaystyle=d^{1-q}c^{-s}d^{-\varepsilon}\left(b^{1-q}c^{-s}b^{-\varepsilon}a^{\varepsilon}c^{s}b^{q}\right)^{\varepsilon}c^{s}d^{q},$
(12) $\displaystyle(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$
$\displaystyle=(W(a,d,c))*_{W,c}(W(b,d,c))$
$\displaystyle=(d^{1-q}c^{-s}d^{-\varepsilon}a^{\varepsilon}c^{s}d^{q})*_{W,c}(d^{1-q}c^{-s}d^{-\varepsilon}b^{\varepsilon}c^{s}d^{q})$
$\displaystyle=W(d^{1-q}c^{-s}d^{-\varepsilon}a^{\varepsilon}c^{s}d^{q},d^{1-q}c^{-s}d^{-\varepsilon}b^{\varepsilon}c^{s}d^{q},c)$
$\displaystyle=\left(d^{1-q}c^{-s}d^{-\varepsilon}b^{\varepsilon}c^{s}d^{q}\right)^{1-q}c^{-s}\left(d^{1-q}c^{-s}d^{-\varepsilon}b^{\varepsilon}c^{s}d^{q}\right)^{-\varepsilon}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\cdot\left(d^{1-q}c^{-s}d^{-\varepsilon}a^{\varepsilon}c^{s}d^{q}\right)^{\varepsilon}c^{s}\left(d^{1-q}c^{-s}d^{-\varepsilon}b^{\varepsilon}c^{s}d^{q}\right)^{q}$
(13)

Consider the case when $\varepsilon=1$ (case 3.2.1). From equalities (12),
(13) we see that the equality
$(a*_{W,c}b)*_{W,c}d=(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$ holds if an only if

$d^{1-q}c^{-s}d^{-1}b^{1-q}c^{-s}b^{-1}ac^{s}b^{q}c^{s}d^{q}=\\\
=\left(d^{1-q}c^{-s}d^{-1}bc^{s}d^{q}\right)^{1-q}c^{-s}d^{-q}c^{-s}b^{-1}ac^{s}d^{q}c^{s}\left(d^{1-q}c^{-s}d^{-}bc^{s}d^{q}\right)^{q}.$
(14)

If $q=1$ (case 3.2.1.1), then equality (14) clearly holds independently on
$s$. It means that in Case 3.2.1.1 we don’t have a quandle. If $q\neq 1$ (case
3.2.1.2), i. e. $q<0$ or $q>1$, then look to $a,b,c,d$ as to the generators of
the free group. If $q<0$, then the reduced word on the left side of equality
(14) ends with $d^{q}$, while the word on the right side of equality (14) ends
with $d^{q-1}$, hence, equality (14) doesn’t hold. If $q>1$, then the reduced
word on the left side of equality (14) begins with $d^{1-q}$, while the word
on the right side of equality (14) begins with $d^{-q}$, hence, equality (14)
doesn’t hold. It means that in Case 3.2.1.2 we don’t have a quandle.

Let us consider the case when $\varepsilon=-1$ (case 3.2.2). From equalities
(12), (13) we see that the equality
$(a*_{W,c}b)*_{W,c}d=(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$ holds if an only if

$d^{1-q}c^{-s}db^{-q}c^{-s}ab^{-1}c^{s}b^{q-1}c^{s}d^{q}=\\\
=\left(d^{1-q}c^{-s}db^{-1}c^{s}d^{q}\right)^{1-q}c^{-s}d^{1-q}c^{-s}db^{-1}ad^{-1}c^{s}d^{q-1}c^{s}\left(d^{1-q}c^{-s}db^{-1}c^{s}d^{q}\right)^{q}.$
(15)

Look to $a,b,c,d$ as to the generators of the free group. If $q<0$, then the
reduced word on the left side of equality (15) ends with $d^{q}$, while the
word on the right side of equality (15) ends with $d^{q-1}$, hence, equality
(15) doesn’t hold. If $q=1$, then using direct calculations it is clear that
equality (15) doesn’t hold. If $q>1$, then the reduced word on the left side
of equality (15) begins with $d^{1-q}$, while the word on the right side of
equality (15) begins with $d^{-q}$, hence, equality (15) doesn’t hold. It
means that in Case 3.2.2 we don’t have a quandle.

Case 4: $|w(y,z)|=3$. In this case $w(y,z)$ is either
$w(y,z)=y^{q_{1}}z^{s}y^{q_{2}}$ for non-zero integers $q_{1},q_{2},s$ or
$w(y,z)=z^{s_{1}}y^{q}z^{s_{2}}$ for non-zero integers $q,s_{1},s_{2}$.

Consider the case when $w(y,z)=y^{q_{1}}z^{s}y^{q_{2}}$ for non-zero integers
$q_{1},q_{2},s$ (case 4.1). In this case
$W(x,y,z)=y\left(y^{q_{1}}z^{s}y^{q_{2}}\right)^{-1}y^{-\varepsilon}x^{\varepsilon}\left(y^{q_{1}}z^{s}y^{q_{2}}\right)=y^{1-q_{2}}z^{-s}y^{-\varepsilon-
q_{1}}x^{\varepsilon}y^{q_{1}}z^{s}y^{q_{2}}$. Let us understand when the
operation $*_{W,c}$ satisfies the third quandle axiom (q3) for each group $G$
and an element $c\in G$

$\displaystyle(a$ $\displaystyle*_{W,c}b)*_{W,c}d=$
$\displaystyle=(W(a,b,c))*_{W,c}d$
$\displaystyle=(b^{1-q_{2}}c^{-s}b^{-\varepsilon-
q_{1}}a^{\varepsilon}b^{q_{1}}c^{s}b^{q_{2}})*_{W,c}d$
$\displaystyle=W\left(b^{1-q_{2}}c^{-s}b^{-\varepsilon-
q_{1}}a^{\varepsilon}b^{q_{1}}c^{s}b^{q_{2}},d,c\right)$
$\displaystyle=d^{1-q_{2}}c^{-s}d^{-\varepsilon-
q_{1}}\left(b^{1-q_{2}}c^{-s}b^{-\varepsilon-
q_{1}}a^{\varepsilon}b^{q_{1}}c^{s}b^{q_{2}}\right)^{\varepsilon}d^{q_{1}}c^{s}d^{q_{2}}$
(16) $\displaystyle(a$ $\displaystyle*_{W,c}d)*_{W,c}(b*_{W,c}d)=$
$\displaystyle=(W(a,d,c))*_{W,c}(W(b,d,c))$
$\displaystyle=\left(d^{1-q_{2}}c^{-s}d^{-\varepsilon-
q_{1}}a^{\varepsilon}d^{q_{1}}c^{s}d^{q_{2}}\right)*_{W,c}\left(d^{1-q_{2}}c^{-s}d^{-\varepsilon-
q_{1}}b^{\varepsilon}d^{q_{1}}c^{s}d^{q_{2}}\right)$
$\displaystyle=W(d^{1-q_{2}}c^{-s}d^{-\varepsilon-
q_{1}}a^{\varepsilon}d^{q_{1}}c^{s}d^{q_{2}},d^{1-q_{2}}c^{-s}d^{-\varepsilon-
q_{1}}b^{\varepsilon}d^{q_{1}}c^{s}d^{q_{2}},c)$
$\displaystyle=\left(d^{1-q_{2}}c^{-s}d^{-\varepsilon-
q_{1}}b^{\varepsilon}d^{q_{1}}c^{s}d^{q_{2}}\right)^{1-q_{2}}c^{-s}\left(d^{1-q_{2}}c^{-s}d^{-\varepsilon-
q_{1}}b^{\varepsilon}d^{q_{1}}c^{s}d^{q_{2}}\right)^{-\varepsilon-q_{1}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \cdot\left(d^{1-q_{2}}c^{-s}d^{-\varepsilon-
q_{1}}a^{\varepsilon}d^{q_{1}}c^{s}d^{q_{2}}\right)^{\varepsilon}\left(d^{1-q_{2}}c^{-s}d^{-\varepsilon-
q_{1}}b^{\varepsilon}d^{q_{1}}c^{s}d^{q_{2}}\right)^{q_{1}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \cdot c^{s}\left(d^{1-q_{2}}c^{-s}d^{-\varepsilon-
q_{1}}b^{\varepsilon}d^{q_{1}}c^{s}d^{q_{2}}\right)^{q_{2}}$ (17)

Consider the case when $\varepsilon=1$ (case 4.1.1). From equalities (16),
(17) we see that the equality
$(a*_{W,c}b)*_{W,c}d=(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$ holds if an only if

$\displaystyle d^{1-q_{2}}c^{-s}$ $\displaystyle
d^{-1-q_{1}}b^{1-q_{2}}c^{-s}b^{-1-q_{1}}ab^{q_{1}}c^{s}b^{q_{2}}d^{q_{1}}c^{s}d^{q_{2}}=$
$\displaystyle=\left(d^{1-q_{2}}c^{-s}d^{-1-q_{1}}bd^{q_{1}}c^{s}d^{q_{2}}\right)^{1-q_{2}}c^{-s}\left(d^{1-q_{2}}c^{-s}d^{-1-q_{1}}bd^{q_{1}}c^{s}d^{q_{2}}\right)^{-1-q_{1}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \cdot
d^{1-q_{2}}c^{-s}d^{-1-q_{1}}ad^{q_{1}}c^{s}d^{q_{2}}\left(d^{1-q_{2}}c^{-s}d^{-1-q_{1}}bd^{q_{1}}c^{s}d^{q_{2}}\right)^{q_{1}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \cdot
c^{s}\left(d^{1-q_{2}}c^{-s}d^{-1-q_{1}}bd^{q_{1}}c^{s}d^{q_{2}}\right)^{q_{2}}$
(18)

If $q_{2}=1$, then equality (18) can be rewritten in the following form

$d^{-1-q_{1}}c^{-s}b^{-1-q_{1}}ab^{q_{1}}c^{s}=\\\
=\left(c^{-s}d^{-1-q_{1}}bd^{q_{1}}c^{s}d\right)^{-1-q_{1}}c^{-s}d^{-1-q_{1}}ad^{q_{1}}c^{s}d\left(c^{-s}d^{-1-q_{1}}bd^{q_{1}}c^{s}d\right)^{q_{1}}d^{-1-q_{1}}$
(19)

If $q_{1}=-1$ (case 4.1.1.1), then equality (19) holds independently on $s$.
It means that in Case 4.1.1.1 we have a quandle. If $q_{1}\neq 1$ (case
4.1.1.2), i. e. $q_{1}<-1$ or $q_{1}>0$, then look to $a,b,c,d$ as to the
generators of the free group. If $q_{1}<-1$, then the reduced word on the left
side of equality (19) end with $c^{s}$, while the word on the right side of
equality (19) end with $d^{-1-q_{1}}$, hence, equality (19) doesn’t hold. If
$q_{1}>0$, then the reduced word on the left side of equality (19) ends with
$c^{s}$, while the word on the right side of equality (19) ends with
$d^{-q_{1}}$, hence, equality (19) doesn’t hold. It means that in Case 4.1.1.2
we don’t have a quandle.

If $q_{2}\neq 1$ (case 4.1.1.3), i. e. $q_{2}<0$ or $q_{2}>1$, then look to
$a,b,c,d$ as to the generators of the free group. If $q_{2}<0$, then the word
on the left side of equality (18) ends with $d^{q_{2}}$, while the word on the
right side of equality (18) ends with $d^{q_{2}-1}$, hence, equality (18)
doesn’t hold. If $q_{2}>1$, then the word on the left side of equality (18)
begins with $d^{1-q_{2}}$, while the word on the right side of equality (18)
begins with $d^{-q_{2}}$, hence, equality (18) doesn’t hold. It means that in
Case 4.1.1.3 we don’t have a quandle.

Codsider the case when $\varepsilon=-1$ (case 4.1.2). From equalities (16),
(17) we see that the equality
$(a*_{W,c}b)*_{W,c}d=(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$ holds if an only if

$\displaystyle d^{1-q_{2}}c^{-s}$ $\displaystyle
d^{1-q_{1}}b^{-q_{2}}c^{-s}b^{-q_{1}}ab^{q_{1}-1}c^{s}b^{q_{2}-1}d^{q_{1}}c^{s}d^{q_{2}}=$
$\displaystyle=\left(d^{1-q_{2}}c^{-s}d^{1-q_{1}}b^{-1}d^{q_{1}}c^{s}d^{q_{2}}\right)^{1-q_{2}}c^{-s}\left(d^{1-q_{2}}c^{-s}d^{1-q_{1}}b^{-1}d^{q_{1}}c^{s}d^{q_{2}}\right)^{1-q_{1}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \cdot
d^{-q_{2}}c^{-s}d^{-q_{1}}ad^{q_{1}-1}c^{s}d^{q_{2}-1}\left(d^{1-q_{2}}c^{-s}d^{1-q_{1}}b^{-1}d^{q_{1}}c^{s}d^{q_{2}}\right)^{q_{1}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \cdot
c^{s}\left(d^{1-q_{2}}c^{-s}d^{1-q_{1}}b^{-1}d^{q_{1}}c^{s}d^{q_{2}}\right)^{q_{2}}$
(20)

If $q_{2}=1$ (case 4.1.2.1), then equality (20) can be rewritten in the
following form

$\displaystyle d^{1-q_{1}}b^{-1}c^{-s}b^{-q_{1}}ab^{q_{1}-1}c^{s}=$
$\displaystyle=\left(c^{-s}d^{1-q_{1}}b^{-1}d^{q_{1}}c^{s}d\right)^{1-q_{1}}d^{-1}c^{-s}d^{-q_{1}}ad^{q_{1}-1}c^{s}\left(c^{-s}d^{1-q_{1}}b^{-1}d^{q_{1}}c^{s}d\right)^{q_{1}}d^{1-q_{1}}b^{-1}$
(21)

If we look to $a,b,c,d$ as to the generators of the free group, then it is
clear that the word on the left side of equality (21) ends with $c^{s}$, while
the word on the right side of equality (21) ends with $b^{-1}$, hence,
equality (21) doesn’t hold. It means that in Case 4.1.2.1 we don’t have a
quandle.

If $q_{2}\neq 1$ (case 4.1.2.2), i. e. $q_{2}<0$ or $q_{2}>1$, then look to
$a,b,c,d$ as to the generators of the free group. If $q_{2}<0$, then the word
on the left side of equality (20) ends with $d^{q_{2}}$, while the word on the
right side of equality (20) ends with $d^{q_{2}-1}$, hence, equality (20)
doesn’t hold. If $q_{2}>1$, then the word on the left side of equality (20)
begins with $d^{1-q_{2}}$, while the word on the right side of equality (20)
begins with $d^{-q_{2}}$, hence, equality (20) doesn’t hold. It means that in
Case 4.1.2.2 we don’t have a quandle.

Let us consider the case when $w(y,z)=z^{s_{1}}y^{q}z^{s_{2}}$ for non-zero
integers $q,s_{1},s_{2}$ (case 4.2). In this case

$W(x,y,z)=y\left(z^{s_{1}}y^{q}z^{s_{2}}\right)^{-1}y^{-\varepsilon}x^{\varepsilon}\left(z^{s_{1}}y^{q}z^{s_{2}}\right)=yz^{-s_{2}}y^{-q}z^{-s_{1}}y^{-\varepsilon}x^{\varepsilon}z^{s_{1}}y^{q}z^{s_{2}}.$
(22)

Let us understand when the operation $*_{W,c}$ satisfies the third quandle
axiom (q3) for each group $G$ and an element $c\in G$

$\displaystyle(a*_{W,c}b)*_{W,c}d$ $\displaystyle=(W(a,b,c))*_{W,c}d$
$\displaystyle=W(W(a,b,c),d,c)$ (23)
$\displaystyle(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$
$\displaystyle=(W(a,d,c))*_{W,c}(W(b,d,c))$
$\displaystyle=W(W(a,d,c),W(b,d,c),c)$ (24)

Let us look to the elements $a,b,c,d$ as to the generators of the free group.
For a word $A\in F(a,b,c,d)$ denote by $|A|_{c}$ the number of syllables which
are the powers of the element $c$ in the reduced form of $A$. Let us evaluate
the values $|(a*_{W,c}b)*_{W,c}d|_{c}$ and
$|(a*_{W,c}d)*_{W,c}(b*_{W,c}d)|_{c}$ given by formulas (23), (24).

First, let us find the upper bound for the value $|(a*_{W,c}b)*_{W,c}d|_{c}$.
Denote by $A_{1}=W(a,b,c)$. From equality (23) and equality (22) it follows
that

$\displaystyle(a*_{W,c}b)*_{W,c}d$
$\displaystyle=W(A,d,c)=dc^{-s_{2}}d^{-q}c^{-s_{1}}d^{-\varepsilon}A_{1}^{\varepsilon}c^{s_{1}}d^{q}c^{s_{2}}$
(25)

In order to calculate the upper bound for $|(a*_{W,c}b)*_{W,c}d|_{c}$ we have
to calculate the number of syllables which are the powers of the element $c$
in the right part of equality (25) without thinking if some powers of $c$ can
be reduced with each other (since we try to calculate the upper bound). We
have the following upper bound

$\displaystyle|(a*_{W,c}b)*_{W,c}d|_{c}\leq 1+1+|A_{1}|+1+1=|A_{1}|+4.$

Let us now calculate the lower bound for the value
$|(a*_{W,c}d)*_{W,c}(b*_{W,c}d)|_{c}$. Denote by $A_{2}=W(a,d,c)$,
$B=W(b,d,c)$. From equality (24) and equality (22) it follows that

$\displaystyle(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$
$\displaystyle=W(A_{2},B,c)=Bc^{-s_{2}}B^{-q}c^{-s_{1}}B^{-\varepsilon}A_{2}^{\varepsilon}c^{s_{1}}B^{q}c^{s_{2}}.$
(26)

In order to calculate the lower bound for
$|(a*_{W,c}d)*_{W,c}(b*_{W,c}d)|_{c}$ we have to calculate the number of
syllables which are powers of $c$ in the right part of equality (26) which are
guaranteed not to be reduced or merged with other syllables which are powers
of $c$.

From the equality $B=W(b,d,c)$ and equality (22) it follows that if $|q|\neq
1$ (case 4.2.1), then in the word $B^{\pm q}$ there are at least $4(q-1)$
syllables which are powers of $c$ which are guaranteed not to be reduced or
merged with other syllables which are powers of $c$ in the right part of
equality (26). This is due to the fact that between $2$ consequent syllables
which are powers of $b$ in the word $B^{\pm q}=W(b,d,c)^{\pm q}$ there are
exactly $4$ syllables which are powers of $c$ which are guaranteed not to be
reduced or merged with other syllables which are powers of $c$, and in the
word $B^{\pm q}=W(b,d,c)^{\pm q}$ there are exactly $q$ syllables which are
powers of $b$, hence, there are at least $4(q-1)$ syllables which are powers
of $c$. Due to this remark in order to calculate the lower bound for
$|(a*_{W,c}d)*_{W,c}(b*_{W,c}d)|_{c}$ let in the right part of equality (26)
calculate only syllables which are powers of $c$ in the word $B^{q}$, $B^{-q}$
and $A_{2}^{\varepsilon}$ (these syllables which are powers of $c$ are
guaranteed not to be reduced or merged with other syllables which are powers
of $c$). We have the following lower bound

$\displaystyle|(a*_{W,c}d)*_{W,c}(b*_{W,c}d)|_{c}\geq
4(q-1)+|A_{2}|_{c}+4(q-1)=|A_{2}|_{c}+8(q-1).$

Hence we have proved that the following inequalities hold

$\displaystyle|(a*_{W,c}b)*_{W,c}d|_{c}$ $\displaystyle\leq|A_{1}|_{c}+4,$
$\displaystyle|(a*_{W,c}d)*_{W,c}(b*_{W,c}d)|_{c}$
$\displaystyle\geq|A_{2}|_{c}+8(q-1).$

Noting that $|A_{1}|_{c}=|W(a,b,c)|_{c}=|W(a,d,c)|_{c}=|A_{2}|_{c}$ we
conclude that

$|(a*_{W,c}b)*_{W,c}d|_{c}\neq|(a*_{W,c}d)*_{W,c}(b*_{W,c}d)|_{c}$

and hence $(a*_{W,c}b)*_{W,c}d\neq(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$, i. e. in
this case there exists a group $G$ and an element $c\in G$ such that
$(G,*_{W,c})$ is not a quandle. It means that in Case 4.2.1 we don’t have a
quandle.

Let us now consider the case when $|q|=1$ (case 4.2.2), i. e. the case when
either $q=1$, or $q=-1$. In this case we can directly calculate all syllabels
which are powers of $c$ in both $(a*_{W,c}b)*_{W,c}d$ and
$(a*_{W,c}d)*_{W,c}(b*_{W,c}d)$, and make sure that
$|(a*_{W,c}b)*_{W,c}d|_{c}\neq|(a*_{W,c}d)*_{W,c}(b*_{W,c}d)|_{c}$. Hence

$(a*_{W,c}b)*_{W,c}d\neq(a*_{W,c}d)*_{W,c}(b*_{W,c}d),$

i. e. in this case there exists a group $G$ and an element $c\in G$ such that
$(G,*_{W,c})$ is not a quandle. It means that in Case 4.2.2 we don’t have a
quandle.

Case 5: $|w(y,z)|\geq 4$. In this case the word $w(y,z)$ has at least two
syllables which are powers of $z$. Using the same technique of evaluating the
values

$\displaystyle|(a*_{W,c}b)*_{W,c}d|_{c}$ and
$\displaystyle|(a*_{W,c}d)*_{W,c}(b*_{W,c}d)|_{c}$

as we did in case 4.2.1, repeating the proof of case 4.2.1 almost word by word
we conclude that

$\displaystyle|(a*_{W,c}b)*_{W,c}d|_{c}<|(a*_{W,c}d)*_{W,c}(b*_{W,c}d)|_{c},$

there exists a group $G$ and an element $c\in G$ such that $(G,*_{W,c})$ is
not a quandle. It means that in Case 5 we don’t have a quandle. $\square$

##  3 Verbal quandles with $n$ parameters

In this short section, we note that there is no sense to classify verbal
quands with more than one parameter. More precisely, using the technique used
for proving Theorem 2, we can show that the following statement about verbal
quands with $n$ parameters holds.

######  Theorem 3.

Let $W\in F(x,y,z_{1},z_{2},\dots,z_{n})$ be such that
$Q=(G,*_{W,c_{1},\dots,c_{n}})$ is a quandle for every group $G$ and elements
$c_{1},c_{2},\dots,c_{n}\in G$. Then $W$ has one of the following forms

  1. 1.

$W=yx^{-1}y$,

  2. 2.

$W=y^{-n}xy^{n}$ for $n\in\mathbb{N}$,

  3. 3.

$W=yu^{-1}xy^{-1}u$ for $u\in F(z_{1},z_{2},\dots,z_{n})$,

  4. 4.

$W=u^{-1}xy^{-1}uy$ for $u\in F(z_{1},z_{2},\dots,z_{n})$,

  5. 5.

$W=yu^{-1}y^{-1}xu$ for $u\in F(z_{1},z_{2},\dots,z_{n})$,

  6. 6.

$W=u^{-1}y^{-1}xuy$ for $u\in F(z_{1},z_{2},\dots,z_{n})$.

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Elizaveta Markhinina
Novosibirsk State University<EMAIL_ADDRESS>
Timur Nasybullov
Novosibirsk State University, Sobolev Institute of Mathematics,
<EMAIL_ADDRESS>