# Holo-VQVAE: VQ-VAE for phase-only holograms

Joohyun Park
KyungHee University
Korea, Republic of
<EMAIL_ADDRESS>Hyeongyeop Kang
KyungHee University
Korea, Republic of
<EMAIL_ADDRESS>
###### Abstract

Holography stands at the forefront of visual technology innovation, offering
immersive, three-dimensional visualizations through the manipulation of light
wave amplitude and phase. Contemporary research in hologram generation has
predominantly focused on image-to-hologram conversion, producing holograms
from existing images. These approaches, while effective, inherently limit the
scope of innovation and creativity in hologram generation. In response to this
limitation, we present Holo-VQVAE, a novel generative framework tailored for
phase-only holograms (POHs). Holo-VQVAE leverages the architecture of Vector
Quantized Variational AutoEncoders, enabling it to learn the complex
distributions of POHs. Furthermore, it integrates the Angular Spectrum Method
into the training process, facilitating learning in the image domain. This
framework allows for the generation of unseen, diverse holographic content
directly from its intricately learned latent space without requiring pre-
existing images. This pioneering work paves the way for groundbreaking
applications and methodologies in holographic content creation, opening a new
era in the exploration of holographic content.

##  1 Introduction

Holography is an emerging technology that unveils a realm where the
conventional boundaries of image capturing and representation are transcended.
Traditional photography is confined to recording the amplitude of light waves,
making it lose three-dimensionality and present a two-dimensional portrayal of
the world. In contrast, holography is able to encapsulate not just the
amplitude, but also the phase values of light waves, thereby preserving the
spatial information intrinsic to every scene.

Unfortunately, the pace of evolution in 3D holography is slower than that in
2D images due to the technical complexity, expensive equipment, computational
intensity, and material limitations [32, 4, 36]. To overcome this, many
researchers begin with simpler forms of holography such as 2D phase-only
holograms (POHs). They conduct experiments and test their hypothesis in the 2D
domain to obtain a foundational understanding of light’s behavior and the
principles of interference and diffraction [26, 19]. The gained understanding
serves as a prerequisite for exploring the more complex domain of 3D
holography. In a similar context, POHs are actively researched because they
simplify the data handling process by keeping the amplitude of the wavefront
constant and only considering the phase variants information to encode the
information of the scene.

These days, the research topic in the image domain has begun to include
generative models for 2D images. Such models have transformative potential in
synthesizing realistic and high-quality visuals, augmenting datasets, and
enhancing various applications from art to healthcare. However, due to its
complexity, only a few attempts have been made to translate this success to
the realm of holography. In terms of phase data learning, generative models
have to process intricate patterns and dependencies between different phases
and phase wraps that can cause abrupt changes and discontinuities in the phase
data. Furthermore, developing appropriate metrics and loss functions that
consider the unique characteristics of phase information becomes a challenge.
These contrast with the simple, pixel-based intensity values commonly
encountered in 2D images, making it challenging to train generative models for
holography.

Previous research of Liu _et al_. [19] demonstrates that a Variational
AutoEncoder(VAE) model trained in the phase domain managed to reconstruct the
input POH but failed to generate novel samples. This suggests that
conventional learning methods face challenges when the model is trained
directly in the phase domain.

In this paper, we introduce Holo-VQVAE, a novel end-to-end generative model
architecture for 2D POH generation. Inspired by HoloNet [26], we find that
using images as input and computing the reconstruction loss in the image
domain allows the model to learn the data distribution and sample novel POHs.
Specifically, we apply the Angular Spectrum Method (ASM) on the generated POH
to reconstruct an image, which is then compared with the input image to
compute the reconstruction loss. Though the results generated using a standard
VAE were promising, we find that adopting the discrete latent space of Vector-
Quantized Variational Autoencoder(VQ-VAE) [40, 28] can further improve the
generation quality. The discrete latent space mitigates the entanglement of
the learned features, resulting in high-quality POH generation.

In summary, the contributions of this paper are as follows:

  * •

We introduce Holo-VQVAE, a pioneer end-to-end generative model architecture
for POHs.

  * •

We facilitate the training process by integrating ASM propagation, enabling
training in the image domain rather than the phase domain, and devised a loss
function balancing noise and quality through an experimental ratio of l2 loss
to perceptual loss.

  * •

We employ the discrete latent space of VQVAE for training the POH with complex
distributions and confirmed through experiments that it shows superior
performance compared to using the continuous latent space of a standard VAE.

##  2 Related works

Holograms are fringe patterns that encapsulate the intricate interplay of
light waves reflected from a scene. The hologram itself is a two-dimensional
surface that contains information about the intensity and phase distribution
of light waves, but the image it produces appears three-dimensional and can be
viewed from different angles.

Holograms can be produced through two primary methods: optical generation and
computational generation, with the latter referred to as Computer-Generated
Holography (CGH). In recent years, CGH has gained increasing prominence over
optical methods, primarily due to the stringent environmental conditions and
precise control required for the generation of optical holograms.

In the idealized conception of CGH, both amplitude and phase play pivotal
roles, collaboratively contributing to the multi-dimensional portrayal of
scenes with enhanced depth and perspective. However, due to the technological
complexities and the economic considerations of spatial light modulators
(SLMs), a substantial corpus of holographic research often focuses on either
amplitude-only holograms or POHs [13].

Among these, POHs have gained more research attention. This inclination is
attributed to the inherent capacity of POHs to yield reconstructions marked by
enhanced brightness and clarity [24, 38]. The process of generating POH is
intricately connected to phase retrieval, a technique that focuses on deducing
the phase of input data from the magnitude of its Fourier transform [31].
Phase retrieval is a complex, non-linear, and ill-posed problem. There are
inherent ambiguities and constraints that make the retrieval of phase
information challenging. For this reason, a plethora of algorithms exist in
the realm of phase retrieval, each tailored to address specific aspects of
these challenges.

Traditionally, algorithmic approaches have long been studied. They are
inherently systematic and are grounded in well-defined mathematical
principles. Algorithmic approaches can indeed be primarily classified into
iterative and non-iterative algorithms. Iterative algorithms for phase
retrieval involve a repetitive process of refinement to achieve an accurate
estimation of the phase [10, 2, 48, 8, 12, 25]. On the other hand, non-
iterative algorithms are characterized by their ability to provide rapid phase
estimations, resulting in significantly reduced computation times [20, 39, 37,
22]. However, they are often hampered by issues related to computational
intensity, convergence challenges, sensitivity to noise, and the expertise
required for implementation.

Recently, the advent and proliferation of deep learning techniques have indeed
instigated notable advancements in the field of POH generation. Deep learning
facilitates the rapid generation of high-quality POHs by learning intricate
patterns of light wave interactions within vast datasets, leading to phase
hologram generation that is not only accurate but also adaptable across
diverse scenarios and conditions [42, 50, 29, 43, 47, 26, 23, 41, 45, 51].

The recent integration of deep learning in POH generation has seen significant
strides, yet the incorporation of generative models has not been as
pronounced. Generative models, characterized by their sophisticated latent
space representations, offer advanced techniques for data manipulation [16,
35, 44, 34]. The integration of these models into POH generation holds the
potential to expand the horizon of possibilities in holography, paving the way
for innovative applications and methodologies.

One relevant study is Liu _et al_. [19] that introduces a generative approach
for hologram generation named channeled variational encoder. This involves an
assumption where the target hologram is modeled as the product of the original
hologram and a specific transformation function. While this method circumvents
the complexities associated with mapping intricate phase data into a latent
space, it inherently reveals a limitation to generating more complex scenes
that cannot be derived simply by applying a transformation function.
Furthermore, augmenting the model’s generative range is intrinsically tied to
the availability of a dataset volume. Our model markedly deviates from Liu’s
paradigm, embodying an enhanced capability to directly synthesize unseen POHs
from a learned latent space.

##  3 Method

Figure 1: Overall architecture of Holo-VQVAE

The overall pipeline of our model, Holo-VQVAE, is illustrated in Figure 1. It
is structured around two principal components: the encoder and the decoder.
During the training phase, the encoder’s role is to convert an input 2D image
into a corresponding representation within the latent space. Subsequently, the
decoder is tasked with the generation of a POH from this latent
representation. The objective function regarding reconstruction is computed in
the image domain, which aims to minimize the discrepancy between the original
input image and the image reconstructed from the POH. The architecture of our
model is based on the work by Esser _et al_. [9], which utilizes a discrete
latent space [40] to effectively encapsulate the distinctive features of the
data. This approach enables a condensed yet richly detailed encoding of the
input, ensuring that the essential characteristics are efficiently represented
within a compact latent space. The details on model configurations are
illustrated in the supplementary material.

###  3.1 Encoding

The input to the encoder is a 2D image $I\in\mathbb{R}^{N\times N\times C}$,
where $N$ denotes the width and height of the image and $C$ specifies its
number of channels. The encoding process encompasses the convolution of input
features and their subsequent downsampling to yield the latent vector
$z\in\mathbb{R}^{n\times n\times H}$, where $n$ denotes its width and height
and $H$ specifies its number of channels.

In the design of our Holo-VQVAE architecture, we have determined to use 2D
images as the training input, with the output being a POH. This decision is
underpinned by two primary considerations.

Firstly, the previous study of Liu _et al_. [19] has reported the significant
challenges associated with direct learning from phase data. The inherent
complexity and high sensitivity to errors in phase information can severely
impede the learning process, often leading to a significant loss in
reconstructed results upon minor inaccuracies. By leveraging the image domain,
we are able to exploit a wide range of well-established learning strategies,
circumventing the complexities inherent to phase data and facilitating a more
stable and effective training process.

Secondly, the scarcity of high-quality POH datasets poses a substantial
barrier [49]. Of course, conversion models are available that can turn
standard image datasets into POH datasets [26, 51]. However, the quality of
the holographic data produced is not always comparable to that of the original
images, and the conversion process can be time-consuming. Maintaining the
quality of training data is crucial, as any reduction in its quality can
negatively impact the performance of the trained model. Hence, we prioritize
using datasets in their original form to guarantee the best possible training
input for our Holo-VQVAE.

###  3.2 Vector quantization

The vector quantization process is a crucial step in VQ-VAEs, distinguishing
them from standard VAEs by quantizing a continuous latent vector $z$ into a
discrete counterpart $\hat{z}$. This is achieved using a codebook consisting
of $K$ representative vectors. For each pixel $z_{uv}$ in the continuous
latent space, where $z_{uv}\in\mathbb{R}^{1\times 1\times H}$ and $u,v\in n$,
we measure its similarity to each codebook vector $w_{k}\in\mathbb{R}^{1\times
1\times H}$, where $k\in K$. The similarity is assessed by computing the
squared Euclidean distance. Then, the codebook vector $w_{k}$ that is most
similar to $z_{uv}$ is then chosen as the quantized counterpart
$\hat{z}_{uv}\in\mathbb{R}^{1\times 1\times H}$. Mathematically, this
selection process is defined as follows:

$\hat{z}_{uv}=\underset{w_{k}}{\mathrm{argmin}}\ \|z_{uv}-w_{k}\|^{2}$ (1)

The quantization process is applied across all pixels, transforming the
continuous latent vector $z$ into its discrete equivalent $\hat{z}$, thereby
facilitating the generation of a discrete latent representation of the input
data.

###  3.3 Decoding and reconstruction

During the decoding stage, the quantized vector $\hat{z}$ is processed through
a series of convolutional layers and upsampling steps to construct a POH
$\phi\in\mathbb{R}^{N\times N\times C}$.

The produced POHs encapsulate phase information in the form of intricate
fringe patterns, which do not present a recognizable depiction of the original
scene. To convert the patterns back into a visual representation of the scene,
a numerical reconstruction process is essential. This typically involves
simulating the wavefront propagation, effectively retracing the path of light
as captured in the hologram.

For wavefront propagation, our model employs ASM [11, 21], a method recognized
for its ability to emulate light wave propagation. The application of ASM to
$\phi$ is defined with the following set of equations, as specified in
Equation 2. These equations model the transformation of the wavefront as it
propagates from the SLM plane, through free space, experiencing diffraction,
and finally arriving at the target plane, where the holographic image is
formed:

$\displaystyle\begin{split}&\leavevmode\resizebox{390.25534pt}{}{$\hat{\phi_{c}}(p_{d})=\iint
F\left(e^{i\phi_{c}}\right)H(f_{x},f_{y},p_{d})e^{i2\pi(f_{x}x+f_{y}y)}df_{x}df_{y},$}\\\
\\\
&\leavevmode\resizebox{390.25534pt}{}{$H(f_{x},f_{y},p_{d})=\begin{cases}e^{i\frac{2\pi}{\lambda_{c}}\sqrt{1-\lambda_{c}^{2}(f_{x}^{2}+f_{y}^{2})}p_{d}},&\text{if}\sqrt{f_{x}^{2}+f_{y}^{2}}<\frac{1}{\lambda_{c}},\\\
0&\text{otherwise}\end{cases}$}\end{split}$ (2)

where $p_{d}$ denotes the propagation distance, ${\phi}_{c}$ denotes the $c\in
C$ channel of ${\phi}$, $\hat{\phi}_{c}$ denotes the propagated wavefront of
${\phi}_{c}$ at distance $p_{d}$, $F$ denotes the Fourier Transform,
$\lambda_{c}$ denotes the wavelength corresponding to channel $c$, $x\in N$
and $y\in N$ denote the spatial coordinates, and $f_{x}$ and $f_{y}$ denote
the spatial frequency components of ${\phi}_{c}$ in the $x$ and $y$
directions. It is important to note that ASM is applied separately to each
color channel, as the wavelength $\lambda_{c}$ varies for different colors in
the light spectrum.

The propagated wavefront $\hat{\phi}$ contains both amplitude and phase
information. Here, the phase information does not directly contribute to the
perceived image but rather to the way light interferes to form that image. In
contrast, the amplitude pattern is what needs to be visualized to represent
the reconstructed image from a hologram. Therefore, it is essential to extract
the amplitude patterns of $\hat{\phi}$. The amplitude extraction is defined by
the following equations:

$\hat{I}=s\cdot|\hat{\phi}|$ (3)

where $\hat{I}$ denotes the reconstructed image and $s$ denotes a
hyperparameter representing a variable for considering the gap between the
output values of the wave propagation operator (ASM) and the desired target
[26].

###  3.4 Objective function

The Holo-VQVAE model’s objective function is articulated through a tripartite
formulation, encompassing reconstruction loss ($L_{recon}$), codebook loss
($L_{codebook}$), and commitment loss ($L_{commit}$), each serving a distinct
purpose in the training process.

The reconstruction loss, $L_{recon}$, quantifies the fidelity of the
reconstructed output $\hat{I}$ relative to the original input $I$. It
comprises of an $l2$ loss and the perceptual loss $L_{per}$ [14] between the
images. The perceptual loss works by feeding both the input and the
reconstructed image into a pretrained VGG16 network to compare their feature
maps extracted at designated layers. The layers for feature extraction are
determined by the target features for comparison. Early layers capture low-
level features like edges and blobs, whereas deeper layers reveal high-level
semantic features [27]. Our model employs features extracted from ReLU 2-2 and
ReLU 4-3 for a comprehensive comparison that encompasses both the low-level
and high-level features.

Our reconstruction loss is computed as a weighted sum of the $l2$ norm and the
perceptual loss $L_{per}$ [14], as delineated in Equation 4:

$L_{recon}=\alpha\cdot\|I-\hat{I}\|_{2}^{2}+\beta\cdot L_{per}(I,\hat{I})$ (4)

where $\alpha$ and $\beta$ are the respective weights assigned to the $l2$
norm and perceptual loss components. We discover that incorporating the
$L_{per}$ can mitigate the noise artifacts within the generated POHs, a
phenomenon further explained upon in subsection 4.1

The codebook loss, $L_{codebook}$, and the commitment loss, $L_{commit}$, are
integral to the VQ-VAE architectures, ensuring the encoder’s outputs maintain
a coherent relationship with the codebook vectors. The $L_{codebook}$, as
specified in Equation 5, updates the codebook vectors to better approximate
the encoder’s outputs, thereby encapsulating the data distribution more
effectively:

$L_{codebook}=\|sg[z]-\hat{z}\|_{2}^{2}$ (5)

where $sg$ denotes the stop-gradient operation that prevents gradient
information from flowing back.

The $L_{commit}$ optimizes the encoder to produce latent vectors that closely
align with the codebook vectors, as expressed in Equation 6. This loss
component restricts the latent space from becoming excessively expansive and
ensures its conformity to a certain embedding:

$L_{commit}=\|z-sg[\hat{z}]\|_{2}^{2}$ (6)

The aggregate objective function of Holo-VQVAE, therefore, is the summation of
these individual losses, formalized in Equation 7:

$L_{total}=L_{recon}+L_{codebook}+L_{commit}$ (7)

###  3.5 Sampling

For a standard VAE, the latent space is designed to follow a continuous
Gaussian distribution, which allows for easy sampling. The generation of novel
data can be achieved by sampling a vector from the Gaussian distribution,
which is then passed to the decoder for reconstruction.

In contrast, VQ-VAEs exploit a discrete latent space where the input to the
decoder is represented by a combination of the codebook vectors. Since it is
not straightforward to sample from this space, an additional model is trained
to sample appropriate sequences of the codebook vectors for the generation of
new samples.

We select PixelSnail [7] as the sampling model for Holo-VQVAE. Other models
including diffusion-based [30] and transformer-based [9] methods may provide
better sampling results, but we leave that to future works. The details on
PixelSnail configurations are illustrated in the supplementary material.

##  4 Experiments

We evaluate our model using datasets of MNIST [18], Fashion-MNIST [46] and
CelebA-HQ [15] datasets. The images are resized to $128\times 128$ for CelebA-
HQ, and $64\times 64$ for MNIST and Fashion-MNIST. The propagation distance
$p_{d}$ is set to 15 mm and 21.5 mm for images with $64\times 64$ and
$128\times 128$ resolutions, respectively. The hyperparameter variable $s$
used in the amplitude extraction is set to $0.95$, following the
implementation of Holonet [26]. The optimizer is set to ADAM with a learning
rate of $0.0002$, and the model was trained on each dataset for $100$ epochs
on a single NVIDIA Geforce RTX 3090 GPU. Further details can be found in the
supplementary materials.

The performance of our model was assessed using Peak Signal-to-Noise Ratio
(PSNR), Structural Similarity Index Measure (SSIM), and Fréchet Inception
Distance (FID) as metrics.

PSNR calculates the ratio between the maximum possible power of a signal (in
this paper, an image) and the power of distorting noise that affects the
quality of its representation. A higher PSNR generally indicates better
reconstruction quality.

SSIM is used to measure the similarity between two images by considering
changes in structural information, luminance, and contrast. SSIM values range
between -1 and 1, where 1 indicates perfect similarity.

FID measures the distribution similarity between the generated images and the
real images. A lower FID score indicates a greater similarity, with a score of
0 being the exact match.

Figure 2: An illustration of example POH reconstructions utilizing diverse
loss functions: (a) employing a model trained solely on $l2$ loss; (b)
employing a model trained on a loss function that integrates $l2$ loss with
perceptual loss in a 9:1 ratio; (c) employing a model trained solely on
perceptual loss. The examples present only a single channel (green) of the
input for better noise visualization.

l2 : perceptual | Metric | MNIST | Fashion-MNIST | CelebA-HQ
---|---|---|---|---
100 : 0 | PSNR | 33.53 | 32.24 | 26.91
SSIM | 0.76 | 0.82 | 0.78
90 : 10 | PSNR | 31.86 | 30.38 | 24.70
SSIM | 0.71 | 0.79 | 0.73
0 : 100 | PSNR | 23.09 | 23.14 | 17.74
SSIM | 0.41 | 0.66 | 0.62

Table 1: The average scores of PSNR and SSIM were assessed for the Holo-VQVAE,
trained with diverse l2: perceptual ratios, using test datasets including
MNIST, Fashion-MNIST, and CelebA-HQ datasets.

###  4.1 Reconstruction loss

In image synthesis, generative models typically incorporate either $l2$ loss
or perceptual loss to enhance reconstruction quality. The implementation of
$l2$ loss aims at minimizing pixel-level discrepancies between the generated
and target images, thereby achieving high fidelity in pixel-value
correspondence. Conversely, perceptual loss [14, 33], aims at enhancing the
perceptual similarity between images, aligning more congruently with human
visual processing mechanisms.

Upon applying the perceptual loss for POH synthesis, we observed that it can
also suppress the generation of noise [5, 6, 3] in the reconstructed images.
When the POH was generated from the model trained only on the $l2$ loss, the
reconstructed image shows noticeable noise, as shown in Figure 2 (a).
Conversely, when the POH was generated from the model trained only on the
perceptual loss, the reconstructed image shows pixel value difference compared
to the input image, as shown in Figure 2 (c). Here, we speculate that
perceptual loss may contribute to regulating and smoothing the noise in the
generation of POH.

The sole use of the perceptual loss, however, results in a noticeable drop in
the average PSNR and SSIM scores on the reconstruction as shown in Table 1. In
light of these insights, we combined $l2$ loss with the perceptual loss in a
9:1 ratio, as demonstrated in Figure 2 (b). This combination was empirically
engineered to harmonize the PSNR and SSIM metrics, thereby optimizing the
reconstruction quality.

###  4.2 Evaluation on reconstruction and sampling

Figure 3: POHs and their reconstructions, generated with Holo-VAE and Holo-
VQVAE, using ground truth (GT) images from MNIST, Fashion-MNIST, and CelebA-HQ
datasets.

Model | Metric | MNIST | Fashion-MNIST | CelebA-HQ
---|---|---|---|---
Holo-VAE | PSNR | 23.89 | 21.75 | 17.63
SSIM | 0.65 | 0.60 | 0.46
Holo-VQVAE | PSNR | 31.86 | 30.38 | 24.70
SSIM | 0.71 | 0.79 | 0.73

Table 2: PSNR and SSIM scores on the reconstructed validation set images for
Holo-VAE and Holo-VQVAE.

We conducted comparative evaluations from both reconstruction and sampling
perspectives to validate whether our architecture design successfully serves
our purpose. Since there are no existing studies with networks that have
successfully achieved our goal of generating POHs, the evaluation focused on
two internally developed approaches: Holo-VAE and Holo-VQVAE.

Holo-VAE refers to our initial architecture, employing a standard VAE
architecture instead of a VQ-VAE. The objective function of this architecture
comprises a reconstruction loss and a KL-divergence loss, which collectively
ensure that the latent space conforms to a Gaussian distribution. Our choice
to begin with the VAE was influenced by its straightforward nature and the
existing foundation of research demonstrating its potential in similar
applications. We hypothesized that employing a VAE, with a carefully devised
reconstruction loss $L_{recon}$, would yield high-fidelity POHs, both in
reconstruction and sampling tasks. Holo-VQVAE refers to our architecture
described in this paper.

The result of reconstructions is shown in Figure 3. While Holo-VAE appears to
capture essential features of the ground truth images and generate plausible
outputs, it fails to capture high-level details and results in blurred images.
We speculate that this could be attributed to VAE’s assumption that the latent
space comprises Gaussian distributions. High-level details often correspond to
less probable regions in a Gaussian distribution, and the VAE’s regularization
(KL-divergence loss) may inadvertently smooth out these details to fit the
Gaussian model. This problem of course can be mitigated by reducing the weight
of the KL-divergence loss. However, this inevitably compromises the sampling
ability [1].

Conversely, the Holo-VQVAE demonstrates a notable proficiency in generating
results that are both clearer and more adept at restoring high-level details.
We speculate that this enhanced performance can be attributed to VQ-VAE’s
utilization of discrete latent space. The discrete representation allows VQ-
VAE to handle complex distributions like POH data and preserve distinct
features that might be averaged out or lost in the continuous latent space of
a VAE [40].

The quantitative evaluation of reconstruction tasks, as indicated by the PSNR
and SSIM scores, is detailed in Table 2. These results demonstrate that Holo-
VQVAE generally outperforms Holo-VAE in terms of reconstruction quality.
Particularly, the enhancement in scores is more pronounced as the dataset
complexity escalates from MNIST to CelebA-HQ.

Subsequently, the result of the sampling task is shown in Figure 4. This task
involves generating new, unseen POHs that are representative of the training
dataset, thereby demonstrating the model’s generative capabilities.

Figure 4: Representative POH samples and their reconstructions of Holo-VAE and
Holo-VQVAE trained on MNIST, Fashion-MNIST, and Celeba-HQ datasets. The R, G,
and B placed on the top-left refer to the RGB channels of the sample POH,
respectively.

Similar to the reconstruction task, Holo-VAE often resulted in images with a
loss of high-level detail and a blurred appearance. In contrast, the Holo-
VQVAE produced images that were clear and rich in high-level details. The
effectiveness of each model in this task is quantitatively assessed using the
FID scores, which measure the similarity between a set of 100 generated
samples and a random selection of 100 images from the validation set. As
indicated in Table 3, the Holo-VQVAE demonstrates superior performance in
terms of FID scores, with the improvement being more pronounced as the dataset
transitions from the less complex MNIST to the more intricate CelebA-HQ. More
samples from Holo-VQVAE can be found in the supplementary material.

Model | Metric | MNIST | Fashion-MNIST | CelebA-HQ
---|---|---|---|---
| Holo-VAE
---
FID | 134.7 | 209.1 | 173.9
Holo-VQVAE | 102.1 | 162.7 | 134.1

Table 3: FID scores on the reconstructions of sample POHs for Holo-VAE and
Holo-VQVAE.

It is important to note that the FID scores for our models are relatively high
when compared to those of recent image generation models. This discrepancy can
be attributed to the inherent challenges in generating high-fidelity POHs, a
task that is fundamentally more complex and nuanced than typical image
generation. The presence of noise, a factor we aimed to mitigate as discussed
in subsection 4.1, also contributes to these higher scores. We anticipate that
ongoing advancements and refinements in our approach will lead to improved FID
scores in the future.

##  5 Limitations and future work

Figure 5: Representative POH samples and their reconstructions of Holo-VQVAE
trained on CelebA-HQ of resolution $256\times 256$. The R, G, and B placed on
the top-left refer to the RGB channels of the sample POH, respectively.

In addressing the limitations and future directions of our research, three key
aspects warrant attention.

Firstly, the resolution of the generated POHs presents a significant
challenge. Our model currently supports resolutions up to $256\times 256$
pixels, as demonstrated in Figure 5. However, for practical applications in
holography, notably higher resolutions are required. This need stems from the
specifications of holographic displays, which typically feature pixel sizes
ranging from 3 to 60 $\mu m$ [17]. Consequently, an image reconstructed at our
model’s highest resolution of $256\times 256$ translates to a physical size of
merely about 1.536 $cm$ in both width and height. This size is considerably
smaller than what is needed for effective visualization in real-world
scenarios, such as in holographic displays or augmented reality devices.
Addressing this limitation involves not only scaling up the resolution but
also ensuring that the increase in resolution does not compromise the quality
of the holographic reconstruction.

Secondly, our model’s current design generates POHs optimized for a fixed
propagation distance $p_{d}$. While this approach ensures quality
reconstructions at this specific distance, it limits the hologram’s
effectiveness for viewers positioned at varying distances. Therefore, it is
crucial to develop methods that allow for variable propagation distances in
the training process. Such an advancement would enable the generation of POHs
that maintain high-quality reconstructions across a range of distances,
significantly enhancing the versatility and applicability of our holographic
models in diverse real-world settings.

Lastly, it is noteworthy that our study did not incorporate experimental
validation using real optical devices for assessing the reconstruction
quality. This decision was grounded as our primary aim was to assess the
algorithmic effectiveness of the learning strategies for high-fidelity POH
generation. Advances in computational holography allow for accurate
predictions and assessments of hologram quality through simulations, aligning
with current research trends where computational validation often precedes
resource-intensive optical testing. However, we acknowledge that future work
involving the practical application of these models would benefit from
experimental validation using optical setups, to fully ascertain their
effectiveness in real-world holographic display scenarios. Therefore, future
work will benefit from optical validation to bridge the gap between
theoretical development and practical application.

##  6 Conclusion

Recent image generation methods leveraging generative models have unlocked a
plethora of creative possibilities, ranging from seamless image interpolation
to intricate text-driven modifications. The cornerstone of these advancements
lies in the effective utilization of latent spaces which distilled the crucial
features of target data.

Despite these advancements in image generation, the field of holography,
particularly a generative model for hologram generation, has yet to fully
embrace the potential of latent spaces. This omission has constrained the
holography domain from accessing the advanced capabilities inherent in
generative models, thereby limiting the scope of innovation and creativity in
hologram generation.

To bridge this gap, we present Holo-VQVAE, a novel framework designed to
integrate the generative models into the domain of hologram generation. As a
pioneering effort, Holo-VQVAE marks a significant stride towards enriching the
holography field with the advanced functionalities of generative models. We
anticipate that this integration will catalyze a wave of novel methodologies,
expanding the horizons of holographic technology. Therefore, the potential of
Holo-VQVAE extends beyond current applications, promising to unlock new realms
of exploration and innovation in the creation and manipulation of holographic
content.

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# Virtual Knot Groups

Heather A. Dye  and  Aaron Kaestner

###### Abstract.

For a knot diagram $K$, the classical knot group $\pi_{1}(K)$ is a free group
modulo relations determined by Wirtinger-type relations on the classical
crossings. The classical knot group is invariant under the Reidemeister moves.
In this paper, we define a set of quotient groups associated to a knot diagram
$K$. These quotient groups are invariant under the Reidemeister moves and the
set includes the extended knot groups defined by Boden et al and Silver and
Williams.

##  1\. Introduction

An oriented virtual knot diagram is a decorated immersion of an oriented
$S^{1}$ into the plane with two types of crossings. Classical crossings are
indicated by over/under markings and virtual crossings are circled. An example
of a virtual knot diagram is shown in figure 1.

Figure 1. Labeled, oriented knot

An oriented virtual knot is an equivalence class of virtual knot diagrams
determined by oriented Reidemeister moves (figures 2, 3 and 4) and an oriented
detour move as in figure 5. An oriented detour moves deletes an oriented arc
of the knot diagram that contains only virtual crossings and redraws the arc
in a new location preserving orientation. Any new double points are virtual
crossings.

$\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,1.13011)\put(0.0,0.0){\includegraphics[width=53.18579pt,page=1]{r1movelhs.pdf}}\end{picture}}\end{array}\leftrightarrow\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,0.83704)\put(0.0,0.0){\includegraphics[width=69.81192pt,page=1]{r1movearhs.pdf}}\end{picture}}\end{array}\leftrightarrow\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,0.8329)\put(0.0,0.0){\includegraphics[width=69.95721pt,page=1]{r1movebrhs.pdf}}\end{picture}}\end{array}$
Figure 2. Oriented Reidemeister I

$\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,0.86731)\put(0.0,0.0){\includegraphics[width=67.14717pt,page=1]{r2amovelhs.pdf}}\end{picture}}\end{array}\leftrightarrow\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,1.53897)\put(0.0,0.0){\includegraphics[width=37.84172pt,page=1]{r2amoverhs.pdf}}\end{picture}}\end{array}$
(a) Oriented Reidemeister II a

$\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,0.87883)\put(0.0,0.0){\includegraphics[width=67.14717pt,page=1]{r2bmovelhs.pdf}}\end{picture}}\end{array}\leftrightarrow\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,1.28916)\put(0.0,0.0){\includegraphics[width=45.77502pt,page=1]{r2bmoverhs.pdf}}\end{picture}}\end{array}$
(b) Contra-oriented Reidemeister II b

Figure 3. Oriented Reidemeister II
$\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,0.99738)\put(0.0,0.0){\includegraphics[width=117.43529pt,page=1]{OR3aLHS.pdf}}\end{picture}}\end{array}\leftrightarrow\begin{array}[]{c}\scalebox{0.5}{
\begin{picture}(1.0,1.03272)\put(0.0,0.0){\includegraphics[width=116.50293pt,page=1]{OR3aRHS.pdf}}\put(-0.25565,0.90262){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\put(-0.03715,0.38434){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\makebox(0.0,0.0)[lb]{\smash{}}}\end{picture}}\end{array}$
Figure 4. Oriented Reidemeister III
$\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,0.79457)\put(0.0,0.0){\includegraphics[width=158.42319pt,page=1]{orienteddetourlhs.pdf}}\end{picture}}\end{array}\leftrightarrow\begin{array}[]{c}\scalebox{0.5}{
\begin{picture}(1.0,0.79405)\put(0.0,0.0){\includegraphics[width=158.42319pt,page=1]{orienteddetourrhs.pdf}}\end{picture}}\end{array}$
Figure 5. Oriented detour move

With orientation, a minimal generating set of Reidemeister moves consists of
two Reidemeister I moves (figure 2), two Reidemeister II moves (figure 3) and
a single Reidemeister III move (figure 4) as shown by Polyak [9] and the
oriented detour move (5). The oriented detour move can be decomposed into
oriented versions of the virtual Reidemeister moves shown in figure 6. The
necessary orientations are analogous to the classical orientations.

$\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,0.64286)\put(0.0,0.0){\includegraphics[width=79.66771pt]{vr1lhs.pdf}}\end{picture}}\end{array}\leftrightarrow\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,0.64286)\put(0.0,0.0){\includegraphics[width=79.66771pt]{vr1rhs.pdf}}\end{picture}}\end{array}$
(a) Virtual I

$\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,0.88111)\put(0.0,0.0){\includegraphics[width=65.21866pt]{vr2lhs.pdf}}\end{picture}}\end{array}\leftrightarrow\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,1.41849)\put(0.0,0.0){\includegraphics[width=41.25412pt]{vr2rhs.pdf}}\end{picture}}\end{array}$
(b) Virtual II

$\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,1.1072)\put(0.0,0.0){\includegraphics[width=52.19499pt]{vr3lhs.pdf}}\end{picture}}\end{array}\leftrightarrow\begin{array}[]{c}\scalebox{0.5}{
\begin{picture}(1.0,1.1072)\put(0.0,0.0){\includegraphics[width=52.19499pt]{vr3rhs.pdf}}\end{picture}}\end{array}$
(c) Virtual III

$\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,1.11181)\put(0.0,0.0){\includegraphics[width=52.22662pt]{vr4lhs.pdf}}\end{picture}}\end{array}\leftrightarrow\begin{array}[]{c}\scalebox{0.5}{
\begin{picture}(1.0,1.11249)\put(0.0,0.0){\includegraphics[width=52.19499pt]{vr4rhs.pdf}}\end{picture}}\end{array}$
(d) Virtual IV

Figure 6. Virtual Reidemeister moves

The Gauss code of a knot diagram records the order of the crossings. To find
the Gauss code of the diagram, we select a base point and traverse the knot in
the direction of orientation and record the label of each crossing as it is
traversed. In figure 1, the Gauss code is

$abcacb.$

The parity of a classical crossing in the knot is determined by whether there
are an even or odd number of labels between the two occurences of the crossing
label in the Gauss code [8], [4]. The crossing has even parity if there is an
even number of terms between the two labels. Otherwise, the crossing has odd
parity. In figure 1, the crossing $a$ has even parity and the crossings $b$
and $c$ are oddly intersticed and have odd parity.

In a diagram with no virtual crossings, all classical crossings have even
parity. The parity of a crossing is unchanged by Reidemeister moves and
virtual Reidemeister moves that do not act on that crossing. Only Reidemeister
I and II moves introduce or remove new classical crossings. A Reidemeister I
move introduces a single crossing with even parity. The Reidemeister II move
introduces either two odd crossings or two even crossings with opposite signs.
The Reidemeister III move does not change the parity of any crossing in the
diagram.

For more information about parity, refer to works on parity by Manturov et al
[6].

###  1.1. Knot Groups

In virtual knot theory, researchers have considered a variety of different
groups associated with virtual knots [1]. These groups are quotients of a free
group obtained by associating a generator with each arc or semi-arc of diagram
and a set of Wirtinger-type relations based on the crossings.

(a) Positive crossing

(b) Negative crossing

(c) Labeled virtual crossing

Figure 7. Labeled signed crossings

In the paper,Virtual knot groups and almost classical knots [1], the virtual
knot group, $VG(K)$, is defined using this method. Positive crossings (as
labeled in figure 7) define the relations: $z=y^{-1}xsys^{-1}$ and
$a=sys^{-1}$. Virtual crossings define the relations $a=qyq^{-1}$ and
$z=qxq^{-1}$. Specializations of this group define many of the groups studied
in virtual knot theory. The quandle group, $QG(K)$, is obtained by letting
$s=1$ [8]. If $q=s$, the welded group, $WG(K)$, (defined by Fenn and Rourke
[5]) is obtained. The extended group $EG(K)$ [10] is obtained by letting
$q=1$. Finally, the fundamental group of a classical knot is obtained by
letting $s=q=1$.

Previously, the authors of this paper incorporated parity information in the
knot group structure in Virtual Parity Alexander Polynomial [3]. This resulted
in a knot group where even positive crossings determined the relations
$z=y^{-1}xsys^{-1}$ and $a=sys^{-1}$. The odd crossings give rise to the
relations $a=\theta y\theta^{-1}$ and $z=\theta x\theta^{-1}$. The virtual
crossings define the relations $a=qyq^{-1}$ and $z=qxq^{-1}$. This quotient
group contains parity information, but at the cost of the information about
the sign of the odd crossings.

In section 2, we define a set of quotient groups associated to a virtual knot.
In section 3, we discuss the structure of this set of knot groups and its
relationship to previously studied knot groups. In section 4, we use this
structure to determine if signed parity information can be incorporated into
the group structure.

##  2\. Knot groups defined via elements of $Aut(F_{2n+j})$

Given a virtual knot diagram $K$, we construct two different quotients of the
free group $F_{2n+j}$ that are invariant under the oriented Reidemeister moves
and the detour move.

Let $n$ be the number of classical crossings in $K$ and choose $j\geq 0$. Then
$K$ has $2n$ semi-arcs that begin and end at classical crossings. Each semi-
arc is associated with one of the generators $a_{1},a_{2},\ldots,a_{2n}$ of
$F_{2n+j}$. Then choose two automorphisms of $F_{2n+j}$, $\phi$ and $\theta$.
We quotient $F_{2n+j}$ by the commutator relations determined by $\theta$ and
$\phi$ and relations determined by the set of classical crossings. In the
second group structure, we let $n$ equal the number of classical and virtual
crossings. We then choose three automorphisms of the free group $F_{2n+j}$. We
quotient $F_{2n+j}$ by commutator relations defined by the automorphisms and
relations determined by the classical and virtual crossings.

Within this set of quotient groups, we recover the knot groups described in
[1]. Within the set of quotient groups defined, we are able to obtain a
variety of algebraic structures that have been motivated by knot diagrams and
the Reidemeister moves.

######  Theorem 1.

Let $K$ be an virtual oriented knot diagram with $n$ classical crossings. Let
$j\geq 0$ be an integer and let $\phi$ and $\theta$ be automorphisms of
$F_{2n+j}$. The quotient group $(K,F_{2n+j},\theta,\phi)$ is determined by the
set of $2n$ relations determined by the classical crossings and the
automorphisms as follows. (See figure 7 for crossing labels.) From positive
crossings, we obtain the relations

$\displaystyle z$ $\displaystyle=y\theta(xa^{-1}),$ $\displaystyle a=\phi(y).$

From negative crossings,

$\displaystyle z$ $\displaystyle=\phi^{-1}(x),$ $\displaystyle
a=\theta^{-1}(z^{-1}y)x.$

We denote these relations as $r_{1},r_{2},\ldots r_{2n}$. Let
$R=\\{r_{1},r_{2},\ldots r_{2n}\\}\cup\\{\phi(\theta(a))=\theta(\phi(a)):a\in
F_{2n+j}\\}.$ Then $(K,F_{2n+j},\phi,\theta)=F_{2n+j}/R$ is invariant under
the oriented Reidemeister moves and detour move.

###### Proof.

Consider a labeled positive crossing as shown in figure 7(a)

$\displaystyle z$ $\displaystyle=y\theta(xa^{-1}),$ $\displaystyle a=\phi(y).$

Using the co-oriented Reidemeister II move (figure 3(a)), we calculate that
negative crossings must define

$\displaystyle z$ $\displaystyle=\phi^{-1}(x),$ $\displaystyle
a=\theta^{-1}(z^{-1}y)x.$

The contra-oriented Reidemeister II move (figure 3(b)) is also invariant under
the relations.

We prove invariance under the oriented Reidemeister move III as shown in
figure 8.

$\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,1.0053)\put(0.0,0.0){\includegraphics[width=181.4561pt,page=1]{R3aLHSv2.pdf}}\end{picture}}\end{array}\leftrightarrow\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,0.92097)\put(0.0,0.0){\includegraphics[width=158.1432pt,page=1]{R3aRHSv2.pdf}}\end{picture}}\end{array}$
Figure 8. Labeled Reidemeister III move

From the left hand side in figure 8, we obtain:

$\displaystyle x$ $\displaystyle=a\theta(rs^{-1}),$ $\displaystyle s$
$\displaystyle=\phi(a),$ $\displaystyle z$ $\displaystyle=b\theta(st^{-1}),$
$\displaystyle t$ $\displaystyle=\phi(b),$ $\displaystyle r$
$\displaystyle=\theta^{-1}(y^{-1}c)t,$ $\displaystyle y$
$\displaystyle=\phi^{-1}(t).$

These equations reduce to

(1) $\displaystyle x$ $\displaystyle=ab^{-1}c\theta(\phi(ba^{-1})),$
$\displaystyle y$ $\displaystyle=b,$ $\displaystyle z$
$\displaystyle=b\theta(\phi(ab^{-1})).$

From the right hand side, we obtain:

$\displaystyle m$ $\displaystyle=p\theta(ay^{-1}),$ $\displaystyle\phi(p)$
$\displaystyle=y$ $\displaystyle q$ $\displaystyle=m\theta(cz^{-1}),$
$\displaystyle\phi(m)$ $\displaystyle=z$ $\displaystyle x$
$\displaystyle=\theta^{-1}(p^{-1}q)b,$ $\displaystyle\phi^{-1}(b)$
$\displaystyle=p$

This set of equations reduces to

(2) $\displaystyle x$ $\displaystyle=ab^{-1}c\phi(\theta(ba^{-1})),$
$\displaystyle y$ $\displaystyle=b,$ $\displaystyle z$
$\displaystyle=b\phi(\theta(ab^{-1})).$

Comparing equations 2 and 2,

$\displaystyle ab^{-1}c\theta(\phi(ba^{-1}))$
$\displaystyle=ab^{-1}c\phi(\theta(ba^{-1})),$ $\displaystyle
b\theta(\phi(ab^{-1}))$ $\displaystyle=b\phi(\theta(ab^{-1})).$

Since the automorphisms $\phi$ and $\theta$ commute, then the relations are
invariant under the generating set of the oriented Reidmeister moves. ∎

Next, we incorporate the virtual crossings into the structure of the quotient
group. In this corollary, the endpoints of a semi-arc are at any crossing,
regardless of type.

######  Corollary 2.

Let $K$ be an oriented knot diagram with $n$ total crossings (both classical
and virtual). Choose $j\geq 0$, an integer. Let $\eta,\phi$, and $\theta$ be
automorphisms of $F_{2n+j}$. The quotient group
$(K,F_{2n+j},\theta,\phi,\eta)$ is obtained by quotienting $F_{2n+j}$ by
relations obtained from crossings and automorphisms. For positive crossings

$\displaystyle z$ $\displaystyle=y\theta(xa^{-1}),$ $\displaystyle a=\phi(y).$

For negative crossings,

$\displaystyle z$ $\displaystyle=\phi^{-1}(x),$ $\displaystyle
a=\theta^{-1}(z^{-1}y)x.$

For virtual crossings,

$\displaystyle a$ $\displaystyle=\eta(y),$ $\displaystyle z$
$\displaystyle=\eta^{-1}(x).$

See figure 7 for crossing labels. We obtain the set of relations:

$R=\\{r_{1},r_{2},\ldots
r_{2n}\\}\cup\\{\phi(\theta(a))=\theta(\phi(a)),\phi(\eta(a)=\eta(\phi(a)),\eta(\theta(a))=\theta(\eta(a)):a\in
F_{2n+j}\\}$

Then the quotient group $(K,F_{2n+j},\eta,\theta,\phi)=F_{2n+j}/R$ is
invariant under the oriented Reidemeister moves and detour move.

###### Proof.

We apply Theorem 1 to obtain invariance under the oriented Reidemeister moves.
For the labeled virtual crossing, we let $a=\eta(y)$ and $z=\eta^{-1}(x)$.
Direct computations verify invariance under the virtual Reidemeister I, II,
and III moves. We focus on the virtual Reidemeister IV move (figure 9).

$\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,1.0053)\put(0.0,0.0){\includegraphics[width=181.4561pt,page=1]{VR4aLHSv2.pdf}}\end{picture}}\end{array}\leftrightarrow\begin{array}[]{c}\scalebox{0.5}{\begin{picture}(1.0,0.92097)\put(0.0,0.0){\includegraphics[width=158.1432pt,page=1]{VR4aRHSv2.pdf}}\end{picture}}\end{array}$
Figure 9. Oriented Virtual Reidemeister IV move

From the left hand side of figure 9, we obtain

$\displaystyle x$ $\displaystyle=a\theta(rs^{-1}),$ $\displaystyle s$
$\displaystyle=\phi(a),$ $\displaystyle y$ $\displaystyle=\eta^{-1}(t),$
$\displaystyle r$ $\displaystyle=\eta(c),$ $\displaystyle z$
$\displaystyle=\eta^{-1}(s),$ $\displaystyle t$ $\displaystyle=\eta(b).$

These equations reduce to

$\displaystyle x$ $\displaystyle=a(\theta(\eta(c)\phi(a^{-1}))),$
$\displaystyle b$ $\displaystyle=y,$ $\displaystyle z$
$\displaystyle=\eta^{-1}(\phi(a)).$

From the right hand side, we obtain

$\displaystyle q$ $\displaystyle=m\theta(cz^{-1}),$ $\displaystyle z$
$\displaystyle=\phi(m),$ $\displaystyle m$ $\displaystyle=\eta^{-1}(a),$
$\displaystyle y$ $\displaystyle=\eta(p).$ $\displaystyle p$
$\displaystyle=\eta^{-1}(b),$ $\displaystyle x$ $\displaystyle=\eta(q).$

These equations reduce to

$\displaystyle x$ $\displaystyle=a\eta(\theta(c\phi(\eta^{-1}(a^{-1})))),$
$\displaystyle b$ $\displaystyle=y,$ $\displaystyle z$
$\displaystyle=\phi(\eta^{-1}(a)).$

Since the automorphisms $\eta,\theta$, and $\phi$ commute then the relations
define a quotient group that is invariant under the oriented classical and
virtual Reidemeister moves. ∎

###  2.1. Examples

We focus on specific examples of Theorem 2. Let $K$ be a virtual knot diagram
with $n$ total crossings.

######  Example 2.1.1.

Let $\phi=\eta=\theta=Id$ and let $j=0$. Then $(K,F_{2n},Id,Id,Id)$ defines a
Wirtinger-type presentation group. For classical knots, this is $\pi_{1}(K)$.

######  Example 2.1.2.

Let $\phi=\eta=Id$, then $(K,F_{2n},Id,\theta,Id)$ defines a quandle for a
fixed $\theta$. In this case, $z=y\theta(xy^{-1})$ and $y=a$. In [7], Joyce
defined a quandle operation on a group $G$ by selecting and automorphism of
$G$ and defining the quandle operation $x\rhd y=s(xy^{-1})y$. Building on this
definition, for a subgroup $H$ of $G$, $G/H$ inherits a quandle structure:
$Hx\rhd Hy=Hs(xy^{-1})y$. This is a homogeneous quandle (for any pair of
elements, $a$ and $b$, there is a quandle homomorphism sending $a$ to $b$).

Further, Joyce proved that

######  Theorem 3 ([7], Theorem 7.1).

Every homogenous quandle has a representation of the form $(G/H,z)$ where $z$
defines an inner automorphism.

and

######  Theorem 4 ([7], Theorem 7.2 ).

Every quandle is representable as $(G/H_{1},H_{2}\ldots,z_{1},z_{2},\ldots).$

######  Example 2.1.3.

Let $\theta=Id$, then $(K,F_{2n},Id,Id,\phi)$ defines a virtual biquandle for
a fixed $\phi$. In this case, $z=yx\phi(y^{-1})$ and $a=\phi(y)$. We recall
the virtual biquandle definition given in [2]. Let $X$ be a set and let
$\Delta:X\rightarrow X\times X$, be defined as $\Delta(x)=(x,x)$. Then let
$B,V:X\times X\rightarrow X\times X$ denote invertible maps satisfying four
axioms:

  1. (1)

$V^{2}=Id:X\times X\rightarrow X\times X,$

  2. (2)

There exist unique invertible maps $S,vS:X\times X\rightarrow X\times X$
satisfying

$S(B_{1}(y,x),y)=(B_{2}(y,x),y)\text{ and }vS(V_{1}(y,x),y)=(V_{2}(y,x),x)$

for all $x,y\in X$,

  3. (3)

$(S^{\pm 1}\circ\Delta)_{j}$ and $(vS^{\pm 1}\circ\Delta)_{j}$ satisfying
$(S\circ\Delta)_{1}=(S\circ\Delta)_{2}$ and
$(vS\circ\Delta)_{1}=(vS\circ\Delta)_{2}$

  4. (4)

$B$ and $V$ satisfy the set theoretic Yang Baxter equations

$\displaystyle(B\times Id)(Id\times B)(B\times Id)=(Id\times B)(B\times
Id)(Id\times B)$ $\displaystyle(V\times Id)(Id\times B)(V\times Id)=(Id\times
V)(B\times Id)(Id\times V)$ $\displaystyle(V\times Id)(Id\times V)(V\times
Id)=(Id\times V)(V\times Id)(Id\times V)$

We note that our proof of Theorem 1 and Corollary 2 demonstrates axiom 4. We
define $S$ and $vS$ as

$\displaystyle S(a,z)=(\phi(z),z^{-1}a\phi(z))\text{ and
}vS(a,z)=(\eta(z),\eta(a)).$

Then, $(S\circ\Delta)_{1}=(S\circ\Delta)_{2}$. Our construction satisfies all
the axioms of the virtual biquandle.

######  Example 2.1.4.

Choose $j=3$. Consider $F_{2n+3}$ with generators $a_{1},a_{2}\ldots
a_{2n},s,t,q$. Let $\phi,\eta,\theta$ be elements of the inner automorphism
group of $F_{2n+3}$ determined by $r,s,t$. We recover the extended virtual
knot group, $VG(K)$ from Boden et al [1] and discussed in section 1.1.

##  3\. The Structure of Theorem 1

We consider the set of groups defined by Theorem 1. In figure 10, we diagram
the possible algebraic structures specialized from $(K,F_{2n+j},\theta,\phi)$.
Let $V(K)$ denote the set of knot structures determined by pairs of
automorphisms of $F_{2n+j}$. Let $I$ denote the identity automorphism on
$F_{2n+j}$. If $\theta=I$, then we recover $B(K)$, a set of groups with a
biquandle structures. If $\phi=I$, then we recover $Q(K)$, a set of groups
with quandle structures. The structures $B(K)$ and $Q(K)$ have been studied in
a variety of papers. Further, letting $\phi=\theta$, we obtain $S(K)$ a set of
group structures where $z=y\theta(x\theta(y^{-1}))$ and $a=\theta(y)$. If
$\phi=\theta^{-1}$ then we obtain the set of group structures, $I(K)$ where
$z=y\theta(x)y^{-1}$ and $a=\theta^{-1}(y)$. The structures in $I(K)$ and
$S(K)$ do not appear to have been studies in the literature. Finally, letting
$\theta=\phi=I$, we obtain the classical fundamental group $\pi_{1}(K)$. This
follows the specialization structure given in Boden et al [1] on a larger
scale.

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10. Structure of $V(G)$

##  4\. Parity Knot Groups

We use this structure to construct a knot group that differentiates between
odd and even crossings and retains sign information. In a Reidemeister III
move, there are either three even crossings or 2 odd crossings and 1 even
crossing. There are three possible versions of the oriented Reidemeister III
move when parity is taken into account.

$\begin{array}[]{c}\scalebox{0.75}{
\begin{picture}(1.0,1.15)\put(0.0,0.0){\includegraphics[width=158.62387pt,page=1]{R3aLHScase1v2.pdf}}\end{picture}}\end{array}\leftrightarrow\begin{array}[]{c}\scalebox{0.75}{
\begin{picture}(1.0,0.96794)\put(0.0,0.0){\includegraphics[width=157.42218pt,page=1]{R3aRHSCase1v2.pdf}}\end{picture}}\end{array}$
Figure 11. Case 1

In the diagrams, we have marked the odd crossings with a star.

To study the relations generally, we let $\mathcal{E}$ and $\mathcal{O}$
define $\theta$ for even and odd crossings respectively. We then use
$\mathsf{e}$ and $\mathsf{o}$ to denote $\phi$ for even and odd crossings
respectively. For simplicity, the bar symbol will be used to indicate
inverses. For example, $\bar{\mathcal{O}}=\mathcal{O}^{-1}$. We derive the
relationships for each case. Note that a Reidemeister III contains only even
crossings or two odd crossings and one even crossing. Hence, from the previous
section, $\mathcal{E}$ and $\mathsf{e}$ commute.

In case 1, we refer to figure 11. From the left hand side, we obtain:

$\displaystyle s$ $\displaystyle=\mathsf{o}(a),$ $\displaystyle x$
$\displaystyle=a\mathcal{O}(rs^{-1}),$ $\displaystyle t$
$\displaystyle=\mathsf{o}(b),$ $\displaystyle z$
$\displaystyle=b\mathcal{O}(st^{-1}),$ $\displaystyle y$
$\displaystyle=\bar{\mathsf{e}}(t),$ $\displaystyle r$
$\displaystyle=\bar{\mathcal{E}}(y^{-1}c)t.$

From the right hand side, we obtain

$\displaystyle p$ $\displaystyle=\bar{\mathsf{e}}(b),$ $\displaystyle x$
$\displaystyle=\bar{\mathcal{E}}(p^{-1}q)b,$ $\displaystyle z$
$\displaystyle=\mathsf{o}(m),$ $\displaystyle q$
$\displaystyle=m\mathcal{O}(cz^{-1}),$ $\displaystyle y$
$\displaystyle=\mathsf{o}(p),$ $\displaystyle m$
$\displaystyle=p\mathcal{O}(ay^{-1}).$

This reduces to

(3) $\displaystyle x$
$\displaystyle=a\mathcal{O}[\bar{\mathcal{E}}(\bar{\mathsf{e}}(\mathsf{o}(b^{-1}))c)\mathsf{o}(ba^{-1})]=\bar{\mathcal{E}}\left[\mathcal{O}[a\mathsf{o}(\bar{\mathsf{e}}(b^{-1}))c\mathsf{o}\left[\mathcal{O}(\mathsf{o}(\bar{\mathsf{e}}(b))a^{-1})\bar{\mathsf{e}}(b^{-1})\right]\right]b,$
(4) $\displaystyle y$
$\displaystyle=\bar{\mathsf{e}}(\mathsf{o}(b))=\mathsf{o}(\bar{\mathsf{e}}(b)),$
(5) $\displaystyle z$
$\displaystyle=b\mathcal{O}(\mathsf{o}(a)\mathsf{o}(b^{-1}))=\mathsf{o}\left(\bar{\mathsf{e}}(b)\mathcal{O}(a\mathsf{o}(\bar{\mathsf{e}}(b)))\right).$

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Figure 12. Case 2

In case 2, we refer to figure 12. From the left hand side, we obtain:

$\displaystyle s$ $\displaystyle=\mathsf{e}(a),$ $\displaystyle x$
$\displaystyle=a\mathcal{E}(rs^{-1}),$ $\displaystyle t$
$\displaystyle=\mathsf{o}(b),$ $\displaystyle z$
$\displaystyle=b\mathcal{O}(st^{-1}),$ $\displaystyle y$
$\displaystyle=\bar{\mathsf{o}}(t),$ $\displaystyle r$
$\displaystyle=\bar{\mathcal{O}}(b^{-1}c)t.$

From the right hand side, we obtain

$\displaystyle p$ $\displaystyle=\bar{\mathsf{o}}(b),$ $\displaystyle x$
$\displaystyle=\bar{\mathcal{O}}(p^{-1}q)b,$ $\displaystyle z$
$\displaystyle=\mathsf{e}(m),$ $\displaystyle q$
$\displaystyle=m\mathcal{E}(cz^{-1}),$ $\displaystyle y$
$\displaystyle=\mathsf{o}(p),$ $\displaystyle m$
$\displaystyle=p\mathcal{O}(ay^{-1}).$

Reducing, we obtain

$\displaystyle x$
$\displaystyle=a\mathcal{E}(\mathcal{O}^{-1}(b^{-1}c)\mathsf{o}(b)\mathsf{e}(a^{-1}))=ab\mathcal{O}^{-1}(\mathcal{E}(c\mathsf{e}(\mathcal{O}(ba^{-1})\bar{\mathsf{o}}(b^{-1}))b,$
$\displaystyle y$ $\displaystyle=b,$ $\displaystyle z$
$\displaystyle=b\mathcal{O}(\mathsf{e}(a)\mathsf{o}(b^{-1}))=\mathsf{e}(\bar{\mathsf{o}}(b)\mathcal{O}(ab^{-1})).$

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Figure 13. Case 3

The third case is in figure 13. From the left hand side,

$\displaystyle s$ $\displaystyle=\mathsf{o}(z),$ $\displaystyle x$
$\displaystyle=a\mathcal{O}(rs^{-1}),$ $\displaystyle t$
$\displaystyle=\mathsf{e}(b),$ $\displaystyle z$
$\displaystyle=b\mathcal{E}(st^{-1}),$ $\displaystyle y$
$\displaystyle=\bar{\mathsf{o}}(t),$ $\displaystyle r$
$\displaystyle=\bar{\mathcal{O}}(y^{-1}c)t.$

From the right hand side

$\displaystyle p$ $\displaystyle=\bar{\mathsf{o}}(b,)$ $\displaystyle x$
$\displaystyle=\bar{\mathcal{O}}(p^{-1}q)b,$ $\displaystyle z$
$\displaystyle=\mathsf{o}(m),$ $\displaystyle q$
$\displaystyle=m\mathcal{O}(cz^{-1}),$ $\displaystyle y$
$\displaystyle=\mathsf{e}(p),$ $\displaystyle m$
$\displaystyle=p\mathcal{E}(ay^{-1}).$

This reduces to

(6) $\displaystyle x$
$\displaystyle=ab^{-1}c\mathcal{O}(\mathsf{e}(b)\mathsf{o}(a^{-1}))=ab^{-1}\bar{\mathcal{E}}\left[\mathcal{O}(c\mathsf{o}(\mathcal{E}(ba^{-1}))b^{-1}))\right]b,$
(7) $\displaystyle y$
$\displaystyle=\bar{\mathsf{o}}(\mathsf{e}(b))=\mathsf{e}(\bar{\mathsf{o}}(b)),$
(8) $\displaystyle z$
$\displaystyle=b\mathcal{E}(\mathsf{o}(a)\mathsf{e}(b^{-1})=b\mathsf{o}(\mathcal{E}(ab^{-1})).$

These relations must be satisfied to construct a parity knot group.

We simplify some of the relations and determine that it is necessary for the
automorphisms to satisfy a set of commutator relations.

From equation 3, we observe that

$\displaystyle
a\mathcal{O}[\bar{\mathcal{E}}(\bar{\mathsf{e}}(\mathsf{o}(b^{-1}))c)\mathsf{o}(ba^{-1})]$
$\displaystyle=\bar{\mathcal{E}}\left[\mathcal{O}[a\mathsf{o}(\bar{\mathsf{e}}(b^{-1}))c\mathsf{o}\left[\mathcal{O}(\mathsf{o}(\bar{\mathsf{e}}(b))a^{-1})\bar{\mathsf{e}}(b^{-1})\right]\right]b.$

Letting $a=b=1$, we obtain
$\mathcal{O}(\bar{\mathcal{E}}(c))=\bar{\mathcal{E}}(\mathcal{O}(c))$. Hence
$\mathcal{E}$ and $\mathcal{O}$ commute.

From equation 4, we observe that
$\bar{\mathsf{e}}(\mathsf{o}(b))=\mathsf{o}(\bar{\mathsf{e}}(b))$, so
$\mathsf{e}$ and $\mathsf{o}$ commute.

Finally, from equation 5

$\displaystyle b\mathcal{O}(\mathsf{o}(a)\mathsf{o}(b^{-1}))$
$\displaystyle=\mathsf{o}\left(\bar{\mathsf{e}}(b)\mathcal{O}(a\mathsf{o}(\bar{\mathsf{e}}(b)))\right).$

Letting $b=1$, the equation reduces to
$\mathcal{O}(\mathsf{o}(a))=\mathsf{o}(\mathcal{O}(a))$ so that $\mathcal{O}$
and $\mathsf{o}$ commute.

In a group structure which preserves the parity and the crossing sign
information, the automorphisms must have the following four commutator
relations:
$[\mathsf{e},\mathcal{E}],[\mathsf{o},\mathcal{O}],[\mathcal{O},\mathcal{E}],$
and $[\mathsf{o},\mathsf{e}]$.

##  5\. Restricting our maps

We prove that both parity and crossing information can not be detected within
the specialized structures.

###  5.1. B(K) parity groups

In this section, we analyze the subsets from the group structure and let
$\mathcal{E}=\mathcal{O}=I$.

We consider equation 8. The simplified form of this equation

(9) $\displaystyle b\mathsf{o}(a)\mathsf{e}(b^{-1})$
$\displaystyle=\mathsf{e}(\bar{\mathsf{o}}(b)).$

From the simplified equation 9, we observe that
$b\mathsf{o}(a)\mathsf{e}(b^{-1})=b\mathsf{o}(ab^{-1})$. As a result,
$\mathsf{o}(b)=\mathsf{e}(b)$ for all $b$. Odd and even crossings are not
distinguished.

###  5.2. S(K) parity groups

Here, $\mathcal{E}=\mathsf{e}$ and $\mathcal{O}=\mathsf{o}$. Note that
$\mathcal{E}$ and $\mathcal{O}$ commute since in the previous section, we
concluded that $\mathsf{o}$ and $\mathsf{e}$ commuted.

We focus on equation 8 and simplify:

(10) $\displaystyle b\mathcal{E}(\mathcal{O}(a))\mathcal{E}^{2}(b^{-1})$
$\displaystyle=b\mathcal{O}(\mathcal{E}(ab^{-1})).$

Since $\mathcal{O}$ and $\mathcal{E}$ commute, observe that
$\mathcal{E}(b)=\mathcal{O}(b)$. This group type does not distinguish
crossings by parity.

###  5.3. I(K) parity groups

In this parity group, we let $\mathcal{O}=\bar{\mathsf{o}}$ and
$\mathcal{E}=\bar{\mathsf{e}}$.

We focus on equation 8 and simplify:

(11) $\displaystyle b\mathcal{E}(\bar{\mathcal{O}}(a))b^{-1}$
$\displaystyle=b\bar{\mathcal{O}}(\mathcal{E}(ab^{-1})).$

Letting $a=1$, we obtain $b\bar{\mathcal{O}}(\mathcal{E}(b^{-1}))=1$ for all
$b$. Hence, $\mathcal{E}=\mathcal{O}$. The odd and even crossings are not
distinguished.

###  5.4. Q(K) parity groups

Here $\mathsf{o}=I=\mathsf{e}$. We focus on equation 6. The expressions reduce
to

(12) $\displaystyle ab^{-1}c\mathcal{O}(ba^{-1})$
$\displaystyle=ab^{-1}\bar{\mathcal{E}}(\mathcal{O}(c\mathcal{E}(ba^{-1})b^{-1}))b.$

Letting $a=b=1$, we observe that $c=\bar{\mathcal{E}}(\mathcal{O}(c))$. Again,
$\mathcal{E}=\mathcal{O}$, so crossings are not distinguished.

######  Theorem 5.

For a virtual knot $K$, the groups $B(K),I(K),S(K),$ and $Q(K)$ cannot detect
both parity and crossing information.

###### Proof.

See equations 9, 11, 10, and 12. ∎

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