# Self-supervised Pseudo-colorizing of Masked Cells

Royden Wagner
Perception Group, Institute of Measurement and Control Systems
Karlsruhe Institute of Technology (KIT)
&Carlos Fernandez Lopez
Institute of Measurement and Control Systems
Karlsruhe Institute of Technology (KIT)
&Christoph Stiller
Institute of Measurement and Control Systems
Karlsruhe Institute of Technology (KIT)
Corresponding author<EMAIL_ADDRESS>
###### Abstract

Self-supervised learning, which is strikingly referred to as the dark matter
of intelligence, is gaining more attention in biomedical applications of deep
learning. In this work, we introduce a novel self-supervision objective for
the analysis of cells in biomedical microscopy images. We propose training
deep learning models to pseudo-colorize masked cells. We use a physics-
informed pseudo-spectral colormap that is well suited for colorizing cell
topology. Our experiments reveal that approximating semantic segmentation by
pseudo-colorization is beneficial for subsequent fine-tuning on cell
detection. Inspired by the recent success of masked image modeling, we
additionally mask out cell parts and train to reconstruct these parts to
further enrich the learned representations. We compare our pre-training method
with self-supervised frameworks including contrastive learning (SimCLR),
masked autoencoders (MAEs), and edge-based self-supervision. We build upon our
previous work and train hybrid models for cell detection, which contain both
convolutional and vision transformer modules. Our pre-training method can
outperform SimCLR, MAE-like masked image modeling, and edge-based self-
supervision when pre-training on a diverse set of six fluorescence microscopy
datasets. Code is available at: https://github.com/roydenwa/cell-centroid-
former

_Keywords_ Self-supervised learning, pseudo-colorization, masked image
modeling, physics-informed, vision transformer, convolutional neural network

##  1 Introduction

The ambitious goal of deep learning research is to develop intelligent
generalist models that can solve a wide variety of tasks. Supervised learning
with massive amounts of labeled data does not scale to the complexity of this
goal. In self-supervised learning, however, supervisory signals are generated
from unlabeled data, making it more scalable. Therefore, self-supervised
learning, which is strikingly referred to as the dark matter of intelligence
(LeCun and Misra, 2021), is gaining more attention in many applications of
deep learning. Especially in biomedical applications, where labeling data
often requires support of trained experts, self-supervised learning can
accelerate research progress and reduce costs (Ghesu et al., 2022). In this
work, we introduce a novel self-supervision objective for the analysis of
cells in biomedical microscopy images. In contrast to recent methods for self-
supervised learning on cell images (e.g., Ciga et al. (2022), Perakis et al.
(2021)), we do not use contrastive learning (Hadsell et al., 2006), but
propose a unique self-supervision objective tailored to cell images. We
propose training deep learning models to pseudo-colorize masked cells. We use
a physics-informed pseudo-spectral colormap that is well suited for colorizing
cell topology. Compared to a spectral colormap, the used colormap has a higher
color variance for low intensity levels. Therefore, this pseudo-spectral
colormap can better highlight cell nuclei and their surroundings in areas of
low intensity and low contrast in microscopy images. Inspired by the recent
success of masked image modeling (e.g., He et al. (2022), Bao et al. (2022)),
we additionally mask out cell parts to increase the complexity of the
objective and further enrich the learned representations. Furthermore, we
build upon our previous work (Wagner and Rohr, 2022a) and train hybrid models
for cell detection, which contain both convolutional and vision transformer
modules. The convolutional modules are used to capture local information,
while the vision transformer modules are used to capture global information.
Overall, the contributions of our work are twofold:

  1. 1.

We propose a novel self-supervision objective for the analysis of biomedical
cell images that combines pseudo-colorizing and masked image modeling.

  2. 2.

We use a recent deep learning model for cell detection, which contains both
convolutional and vision transformer modules, to evaluate the proposed self-
supervision objective.

##  2 Related Work

Self-supervised learning on biomedical cell images. Ciga et al. (2022) and
Perakis et al. (2021) use contrastive learning and build upon SimCLR (Chen et
al., 2020) to train deep learning models in a self-supervised manner on cell
images. In contrastive learning for visual representations, the learning
objective is to maximize agreement between two different augmented views of a
single sample, while using the remaining samples in a batch as negative
examples. Experiments show that large batch sizes and correspondingly huge
datasets and computational resources are needed to maximize performance (e.g.,
a batch size of 32k for SimCLR on ImageNet). Dmitrenko et al. (2022) show that
a small convolutional autoencoder (only 190k parameters) trained on cell
images can outperform much larger general purpose pre-trained models (e.g.,
ViT-B/8 (Caron et al., 2021), 85M parameters) on classifying drug effects.
This demonstrates the potential of autoencoding as pre-training on cell
images. However, recent work on autoencoders (e.g., He et al. (2022), Gao et
al. (2022)) suggests that masking parts of the input further enriches the
learned representations. Kobayashi et al. (2022) propose protein
identification as self-supervision objective for pre-training on biomedical
cell images. While being well tailored to cell images, this approach requires
basic annotations of protein IDs. Dawoud et al. (2022) use edge detection as
self-supervision objective to pre-train deep learning models for cell
segmentation. Edge detection is closely related to accurate segmentation of
cell borders, but is a fairly simple task which could lead to less expressive
learned features.
Colorizing as self-supervision objective. Zhang et al. (2016) convert natural
color images to grayscale and use the recolorization to the CIE lab color
space as self-supervision objective. Vondrick et al. (2018) leverage the
temporal coherency of color in natural videos to learn colorizing following
frames based on a reference frame. Both approaches are promising for natural
images or videos, but exploit color-related features that are not present in
biomedical grayscale microscopy images.

##  3 Method

###  3.1 Pseudo-colorize Masked Cells as Self-supervision Objective

Figure 1: (a) Pseudo-colorization of fluorescence microscopy images and the
corresponding colormaps. (b) Masking schemes and masked fluorescence
microscopy images. MAE (He et al., 2022) masks cover 75% of images, whereas
our proposed padded masks contain smaller patches and cover 33%. Image areas
masked by our padded masking scheme are highlighted in white here to enhance
their visibility. During pre-training, these areas are set to zero. (c)
Proposed pre-training objective: Pseudo-colorize masked cells.

Autoencoders (Kramer, 1991) are classical deep learning models for self-
supervised representation learning. During training an autoencoder learns to
map its input to a latent representation and to reconstruct the input from the
latent representation. The general architecture of an autoencoder contains a
contracting part called encoder and an expanding part for reconstruction
called decoder. Popular applications include compression, where the learned
latent representation is smaller than the input, or denoising (Vincent et al.,
2008), where input signals are corrupted by noise and uncorrupted signals are
reconstructed. In this work, we use a new form of autoencoding by training to
reconstruct pseudo-colorized versions of input images. Specifically, we train
deep learning models to pseudo-colorize cell images as self-supervised pre-
training for cell detection. As previously shown (Dmitrenko et al., 2022),
basic autoencoding of cell images can improve the performance on downstream
tasks such as drug effect classification. We argue that reconstruting a
pseudo-colorized version is more difficult than reconstructing the original
input image. Thus, it can lead to more expressive learned features and further
improve the performance on downstream tasks. Figure 1 (a) shows pseudo-
colorized cell images and the characteristics of the used colormaps. We are
using colormaps provided by the matplotlib library (Barrett et al., 2005). The
grayscale plots of the colormaps are generated by computing the perceived
brightness $B_{P}$ from RGB values using the HSP color system (Finley, 2006),
as follows:

$B_{P}=\sqrt{0.299R^{2}+0.587G^{2}+0.114B^{2}}$ (1)

We select four colormaps from different colormap categories to cover a wide
range of colormaps in the following. The rainbow colormap is a physics-
informed spectral colormap based on the visible spectrum of light. The lowest
intensity levels are mapped to violette, the highest intensity levels to red.
The perceived brightness gradually decreases from medium intensity levels
towards low and high intensity levels. The seismic colormap is a monotonically
diverging colormap composed of two colors, blue and red. The perceived
brightness rapidly decreases from medium intensity levels towards low and high
intensity levels. The nipy_spectral colormap is a physics-informed pseudo-
spectral color map that extends spectral colormaps by prepending black for low
intensity levels and appending gray for high intensity levels. Therefore, this
colormap has a higher color variance than spectral colormaps. Furthermore, the
color transitions in the colormap are sharper than in spectral colormaps. This
is further expressed in a less uniform perceived brightness. viridis is a
perceptually uniform sequential colormap composed of violette, blue, green,
and yellow. It contains smooth color transitions and the perceived brightness
is gradually increasing from low intensity levels towards high intensity
levels.

We propose to choose colormaps for pseudo-color autoencoding, where the
generated color channels approximately match the semantics of the data. We
hypothesise that approximating semantic segmentation by pseudo-colorization is
beneficial for self-supervised pre-training on images. For data such as
fluorescence microscopy images, where semantics are closely related to
grayscale intensity levels, spectral or pseudo-spectral colormaps are
suitable. Such colormaps map low intensities primarily to bluish colors,
medium intensities to greenish colors and high intensities to reddish colors.
Thus, the intensity levels are roughly divided into the three color channels
in the RGB scheme and semantic segmentation is approximated. As shown in
Figure 1 (a), the semantic cell mask covers primarily areas of low intensity.
Therefore, the pseudo-spectral nipy_spetral colormap, which has a higher color
variance for such intensity levels, can better highlight cells in this context
than the spectral rainbow colormap. In detail, cell bodies are primarily
colorized in violette, which maps to the blue and red RGB channels, and cell
nuclei are primarily colorized in blue and green.

Recent work on autoencoding as pre-training on images (e.g., He et al. (2022),
Bao et al. (2022), Gao et al. (2022)), suggests that masked image modeling can
lead to more expressive learned features than the classical reconstruction
objective. In masked image modeling, images are divided into non-overlapping
patches and during pre-training these patches are randomly masked out.
Thereby, the objective for an autoencoder becomes to reconstruct seen and
unseen parts of an image. The idea is that during pre-training an autoencoder
builds up internal representations for certain object classes and learns to
leverage these to reconstruct masked image parts. Masked autoencoder (MAE) (He
et al., 2022) is an autoencoder model with an encoder and decoder based on the
vision transformer architecture (Dosovitskiy et al., 2021). Therefore, images
of size $224\times 224$ pixels are divided into non-overlapping $16\times 16$
pixel patches. For an image size of $384\times 384$ pixels, this corresponds
approximately to a patch size of $27\times 27$ pixels. During pre-training,
75% of these patches are randomly masked out. We argue that such a masking
scheme is primarily designed for data where the objective is to identify one
concept per image and not to localize multiple objects within an image. As
shown in Figure 1 (b), the MAE-like masking would mask out a large number of
cells in a Fluo-N2DH-SIM+ microscopy image. Therefore, an autoencoder can not
use partially visible cell parts as clues to reconstruct adjacent masked
patches, but rather has to learn a representation for all cells combined in an
image to reconstruct masked patches. In Figure 1, the MAE-like masking is
therefore applied to a Fluo-C3DH-A549-SIM microscopy image, which only
contains one cell. We argue that learning to leverage local or semi-local
visual clues is beneficial for downstream tasks such as object detection or
segmentation. Thus, we propose an alternate masking scheme for cell detection.
Our padded masking scheme is composed of smaller patches ($12\times 12$ pixels
for an image size of $384\times 384$ pixels) and all patches are padded with a
padding size of $1/4$ of the patch width. This surrounds all patches with a
non-masked area and prevents large connected masked areas, which could mask
out whole cells. The patches are masked out randomly with a probability of 50%
using a discrete uniform distribution $U\\{0,1\\}$ to choose the common pixel
value of a patch. Considering the non-masked padding areas the overall mean
masking ratio becomes $33.\bar{3}\%$. In the following we combine the pseudo-
color autoencoding with masked inputs and thus train deep learning models to
pseudo-colorize masked cells.

###  3.2 Model Architecture

The comparison of vision transformers (ViTs) (Dosovitskiy et al., 2021) and
convolutional neural networks (CNNs) (LeCun et al., 1989) in computer vision
applications reveals that their receptive fields are fundamentally different
(Raghu et al., 2021). The receptive fields of ViTs capture local and global
information in both earlier and later layers. The receptive fields of CNNs, on
the other hand, initially capture local information and gradually grow to
capture global information in later layers. Therefore, we use MobileViT blocks
(Mehta and Rastegari, 2022) in the neck part of our proposed model to enhance
global information compared to a fully convolutional neck part. Figure 2 (a)
shows the proposed model architecture.

Figure 2: (a) CellCentroidFormer model. Backbone: Five blocks of an
EfficientNetV2S (Tan and Le, 2021). Neck: MobileViT blocks and convolutional
layers. Heads: Fully convolutional upsampling blocks. (b) Adjacent z-slices as
input for the ContextBlock.

We represent cells by their centroid, their width, and their height. Our model
contains two fully convolutional heads to predict these cell properties. The
first head predicts a heatmap for cell centroids, and the second head predicts
the cell dimensions (width and height) at the position of the corresponding
cell centroid. The heatmaps for cell centroids are generated by first creating
a semantic map where cells are approximated by ellipses. Afterwards, we smooth
each cell ellipse using a normalized box filter with a kernel size $k$ that is
scaled to the corresponding cell height $h$ and width $w$:

$k=(w//1.5,h//1.5),$ (2)

where $//$ represents integer division. Cell height and width are encoded in
rectangles sized to 50% of the corresponding cell dimensions. These rectangles
contain height and width values scaled relative to the input image size of
$384\times 384$ pixels.

####  3.2.1 Prepended ContextBlock

The ContextBlock is a proposed extension to our CellCentroidFormer model. It
serves two purposes: first, to encode spatial context from adjacent slices
when analyzing 3D inputs slice-wise. Second, to provide our model with a
unified input form of 3 views for 2D and 3D input data to jointly train on
both types of datasets. As previously shown (e.g., Arbelle et al. (2022),
Wagner and Rohr (2022b)), using tripletts of adjacent slices when analyzing 3D
microscopy images of cells slice-wise can improve cell detection and
segmentation. Figure 2 (b) shows how we decompose a 3D microscopy images into
tripletts of adjacent slices. In the ContextBlock, highlevel features are
first extracted from adjacent slices $S$ using convolutional layers. These
features are then merged with the current slice using a multiply-accumulate
operation to generate a context tensor $T_{context}$ with three channels.

$T_{context}(z)=\begin{bmatrix}\sigma(\mathrm{Conv1x1}(\mathrm{Conv3x3}(S(z-1)))\cdot
S(z)+S(z))\\\ S(z)\\\ \sigma(\mathrm{Conv1x1}(\mathrm{Conv3x3}(S(z+1)))\cdot
S(z)+S(z))\end{bmatrix},$ (3)

where $\mathrm{Conv3x3}$ and $\mathrm{Conv1x1}$ represent 2D convolutional
layers with kernel sizes of $3\times 3$ and $1\times 1$ and $\sigma$
represents a sigmoid activation function. A $\mathrm{Conv3x3}$ layer with 10
filters is used to detect high-level features, afterwards these features are
fused along the depth axis using a $\mathrm{Conv1x1}$ layer with one filter.
We merge context from the previous slice $S(z-1)$ in the first channel, the
second channel contains the current slice $S(z)$, and context from the next
slice $S(z+1)$ is merged in the third channel. In this way, context from
adjacent slices can be highlighted, but the focus remains on the current
slice. In general, prepending the ContextBlock is a learned 2.5D approach for
3D data. For 2D data, additional features are extracted and 3 unique views of
input images are generated.

####  3.2.2 Decoding Predictions

When decoding predictions, we first generate a binary map $E_{blob}$ that
contains elliptical blobs by applying a threshold on the centroid heatmap
$H_{centroid}$, as follows:

$E_{blob}(x,y)=H_{centroid}(x,y)>0.75$ (4)

Afterwards, we compute the image moments $M$ per blob and derive the
corresponding centroid position $C$ using:

$C\\{x,y\\}=\left\\{\frac{M_{10}}{M_{00}},\frac{M_{01}}{M_{00}}\right\\}$ (5)

These centroid positions are then used to lookup the corresponding cell
dimensions in the predicted cell height and cell width maps. In this work, we
use the classic bounding box format of top left corner plus height and width
as output.

##  4 Results

###  4.1 Datasets

We use publicly available datasets from the Cell Tracking Challenge (Ulman et
al., 2017) to evaluate our proposed method. In the first set of experiments,
we are using the Fluo-N2DH-SIM+ dataset (Svoboda and Ulman, 2016), which
contains fluorescence microscopy images of simulated nuclei of HL60 cells.
Figure 1 shows an example image of this dataset. In the second set of
experiments, we combine the Fluo-N2DH-SIM+ with the Fluo-C3DH-A549-SIM
(Sorokin et al., 2018), the Fluo-C2DL-Huh7 (Ruggieri et al., 2012), the
Fluo-C3DL-MDA231, the Fluo-N2DH-GOWT1 (Bártová et al., 2011), and the
Fluo-C2DL-MSC datasets for pre-training. The Fluo-C3DH-A549-SIM contains 3D
fluorescence microscopy images of simulated GFP-actin-stained A549 lung cancer
cells. The Fluo-C2DL-Huh7 dataset contains 2D fluorescence microscopy images
of human hepatocarcinoma-derived cells expressing the fusion protein YFP-
TIA-1. The Fluo-C3DL-MDA231 dataset contains 3D fluorescence microscopy images
of human breast carcinoma cells infected with a pMSCV vector including the GFP
sequence, which are embedded in a collagen matrix. The Fluo-N2DH-GOWT1 dataset
contains 2D fluorescence microscopy images of GFP-GOWT1 mouse stem cells. The
Fluo-C2DL-MSC dataset contains 2D fluorescence microscopy images of rat
mesenchymal stem cells on a flat polyacrylamide substrate. Table 2 contains
example images for all used datasets (2D slices for 3D datasets), in the order
in which they are listed here. As pre-processing, we apply a median filter
with a kernel size of 3 and perform min-max scaling. During training, we use
the Fluo-N2DH-SIM+ and Fluo-C3DH-A549-SIM datasets that provide panoptic
segmentation masks as ground truth, which we convert to bounding box
annotations for cells.

###  4.2 Comparing Autoencoding Schemes Including Pseudo-colorization and
Masking as Pre-training on Cell Images

In this set of experiments, we are comparing different autoencoding schemes as
pre-training for cell detection. We perform two types of training procedures,
fine-tuning and head evaluation. For fine-tuning, we pre-train the models
using different autoencoding schemes and afterwards allow all model parameters
to adapt to the downstream task during training. For head evaluation, we pre-
train the models using different autoencoding schemes and afterwards freeze
the weights in the backbone and neck part of our model during training.
Therefore, only the heads are adapting to the downstream task, which
corresponds to 17% of the total model parameters.

Table 1: Comparing autoencoding schemes including pseudo-colorization and
masking as pre-training on cell images. As baseline, the first row shows a
model trained from scratch with random weight initialization. Masking refers
to our proposes padded masking scheme. Best scores per training procedure are
bold, second best scores are underlined.

Pre-training | Training |
---|---|---
Colormap | Masking | SSIM${}_{test}^{heads}$ | FDS | APtest | AP${}_{test}^{50}$ | AP${}_{test}^{small}$ | AP${}_{test}^{medium}$ | mIoU${}^{heatmap}_{train}$ | Trel
- | - | - | 100% | 31.37 (1.27) | 71.93 (0.90) | 27.43 (1.08) | 44.37 (2.06) | 82.78 (0.82) | 1.0
Fine-tuning: |  |  |  |  |  |  |  |  |
- | - | 0.9212 (0.0009) | 100% | 38.93 (2.60) | 78.87 (2.20) | 34.17 (2.40) | 54.33 (3.25) | 90.02 (2.04) | 1.9
rainbow | - | 0.8392 (0.0003) | 100% | 40.60 (1.97) | 79.77 (1.00) | 35.53 (1.81) | 56.53 (2.65) | 89.79 (1.27) | 2.3
viridis | - | 0.8790 (0.0087) | 100% | 40.70 (1.22) | 80.03 (0.85) | 35.97 (1.10) | 56.07 (1.37) |  91.21 (0.11) | 2.3
seismic | - | 0.9020 (0.0035) | 100% | 40.33 (0.49) | 79.80 (0.35) | 35.73 (0.40) | 55.13 (1.46) | 90.41 (0.68) | 2.3
nipy_spectral | - | 0.6302 (0.0004) | 100% |  43.10 (1.51) |  81.63 (0.90) |  38.07 (1.02) |  58.93 (2.93) | 90.95 (0.50) | 2.3
nipy_spectral | ✓ | 0.6861 (0.0004) | 100% |  43.43 (1.90) |  81.90 (1.61) |  38.40 (2.15) |  59.27 (1.30) |  91.30 (0.55) | 2.3
Head evaluation: |  |  |  |  |  |  |  |  |
- | - | 0.9212 (0.0009) | 100% | 22.30 (2.69) | 58.50 (4.77) | 19.30 (2.60) | 32.67 (2.57) | 69.74 (0.86) | 1.6
nipy_spectral | - | 0.6302 (0.0004) | 100% |  30.87 (4.74) |  69.40 (4.42) |  27.30 (3.81) |  43.00 (6.81) |  73.41 (0.69) | 2.1
nipy_spectral | ✓ | 0.6861 (0.0004) | 100% |  32.57 (1.88) |  70.17 (2.77) |  28.30 (1.66) |  47.17 (2.33) |  74.06 (0.34) | 2.1
Head evaluation with smaller training dataset: |  |  |  |  |  |
- | - | 0.9212 (0.0009) | 20% |  9.45 (2.25) | 30.02 (5.91) |  7.00 (1.76) | 18.65 (3.39) | 66.69 (0.83) | 1.1
nipy_spectral | - | 0.6302 (0.0004) | 20% |  16.80 (2.77) |  49.07 (5.00) |  14.57 (2.36) |  24.97 (4.26) |  72.29 (0.62) | 1.5
nipy_spectral | ✓ | 0.6861 (0.0004) | 20% |  15.88 (3.47) |  44.65 (7.92) |  13.28 (3.08) |  25.78 (5.02) |  70.77 (1.17) | 1.5

Dataset. As dataset, we are using the training split of the Fluo-N2DH-SIM+
dataset. We perform geometric data augmentations such as elastic, perspective,
shift, scale, and rotation transformations to increase the dataset size to
2150 samples. 80% of the resulting dataset are used for training and 20% for
testing. The used dataset is rather small and we are comparing similar
autoencoding schemes in this set of experiments. Therefore, we perform data
augmentations beforehand instead of randomly and on-the-fly during training to
train all models with the exact same samples and allow a fair comparison.

Evaluation metrics. As evaluation metrics, we use the structural similarity
score (SSIM) (Wang et al., 2004), the mean intersection-over-union metric
(mIoU), and bounding box average precision metrics (AP). The structural
similarity score measures the similarity of two matrices or grayscale images
and is used to measure the reconstruction quality during pre-training. For
pseudo-colorized images we compute the SSIM score per color channel and
average them. We do not use weight-sharing, but train both heads separately
during pre-training. Therefore, we report the mean SSIM score of both model
heads. The metric is defined as follows:

$\mathrm{SSIM}(x_{1},x_{2})=\frac{(2\mu_{x_{1}}\mu_{x_{2}}+(0.01L)^{2})(2\sigma_{x_{1}x_{2}}+(0.03L)^{2})}{(\mu_{x_{1}}^{2}+\mu_{x_{2}}^{2}+(0.01L)^{2})(\sigma_{x_{1}}^{2}+\sigma_{x_{2}}^{2}+(0.03L)^{2})}$
(6)

For two inputs, the mean $\mu$, the variance $\sigma$, and the dynamic range
$L$ are computed. The mIoU metric is used to evaluate the centroid heatmap
predictions. Matching our decoding procedure in Equation 4, we apply a
threshold to the predicted and ground truth heatmap before computing the mIoU
score. The metric is defined as:

$\mathrm{mIoU}=\frac{1}{C}\sum_{C}\frac{TP_{C}}{TP_{C}+FP_{C}+FN_{C}}$ (7)

For the two class labels $C$, background and cell centroid blobs, the true
positive $TP$, false positive $FP$, and false negative $FN$ pixels are
computed. The bounding box AP metrics are computed using the pycocotools
software library, we refer to the COCO dataset (Lin et al., 2014) for more
details.

Experimental setup. During pre-training, Adam (Kingma and Ba, 2015) with its
standard configuration is used as optimizer, the initial learning rate is set
to $1^{-4}$, and reduced on plateaus by a factor of 10 to a minimum learning
rate of $1^{-6}$. As pre-training loss, we are computing the pixel-wise mean
squared error (MSE) between the reconstructed and the original microscopy
images. All autoencoding schemes including pseudo-colorization are pre-trained
for 75 epochs, whereas the basic autoencoding without pseudo-colorization is
pre-trained for 50 epochs since it converges faster. Therefore, the basic
autoencoding scheme has a lower relative total training time $T_{rel}$ than
the remaining autoencoding schemes. During training, the same optimizer
configuration and learning rate scheduling are used. As training loss, three
Huber loss functions (Huber, 1964) are used, one loss function per output
(heatmap, height, and width). The total loss is computed by a weighted sum of
the three loss values:

$\mathcal{L}_{Huber}(y,\widehat{y})=\begin{cases}\frac{1}{2}(y-\widehat{y})^{2}&\text{if
}y-\widehat{y}\leq 1.0,\\\ (y-\widehat{y})-\frac{1}{2}&\text{else}\end{cases}$
(8)
$\mathcal{L}_{total}=\mathcal{L}_{heatmap}+\frac{1}{2}\cdot\mathcal{L}_{height}+\frac{1}{2}\cdot\mathcal{L}_{width}$
(9)

All models are trained for 50 epochs to detect cells. All training runs are
completed three times, Table LABEL:tab:tab1 shows the mean and standard
deviation for all considered metrics.

Results. Basic autoencoding without pseudo-colorization yields the highest
SSIM scores during pre-training, autoencoding using the viridis and seismic
colormaps yields the second highest SSIM scores. This shows that these pre-
training objectives are simpler than the remaining ones and therefore easier
to learn. More difficult to learn seems autoencoding using the spectral
rainbow colormap and most difficult are the two variations with and without
masking using the pseudo-spectral nipy_spectral colormap. For all autoencoding
schemes, the standard deviation from the SSIM score are low, showing that both
model heads learn the objectives similarly well. In the fine-tuning training
procedure, all autoencoding schemes achieve at least 10% higher values in all
metrics than the baseline model trained from scratch. This demonstrates that
autoencoding is generally an effective form of pre-training for this type of
data. All cells in the Fluo-N3DH-SIM+ dataset fall into the small and medium
large object categories with respect to COCO AP metrics. In general, medium
sized cells are better detected, which is shown by higher AP scores. The basic
autoencoding achieves on average about 2% lower AP scores than the very
similar performing autoencoding schemes with the rainbow, viridis, and seismic
colormaps. The two autoencoding schemes with the nipy_spectral colormap
achieve AP scores that are about another 2.5% higher. When comparing these two
autoencoding schemes, the autoencoding scheme that uses the proposed padded
masking in addition to the nipy_spectral colormap achieves AP values that are
consistently about 0.3% higher and thus performs best. The mIoU scores, which
are achieved during training, are close to each other in this training
procedure and within a range of 1.5%. In the following head evaluation
procedure, the two previously best performing autoencoding schemes are
compared with the basic autoencoding scheme. According to the significantly
lower number of parameters, which can adapt to the downstream task, the
performance drops significantly. On average, 10 to 20% lower AP scores are
achieved. The performance differences between the three autoencoding schemes
are comparable to those from the previous procedure. Therefore, the
autoencoding scheme using the combination of nipy_spectral colormap and padded
masking again performs best. This autoencoding scheme is also the only one
that outperforms the differently trained baseline model with this training
procedure. We increase the difficulty further by reducing the fraction of the
training dataset $F_{DS}$ used to 20%. With this setup, the achieved AP scores
are approximately halved. The two autoencoding schemes with the nipy_spectral
colormap outperform the basic autoencoding again, whereas the autoencoding
scheme with the nipy_spectral colormap but without masking performs best in
this setup. We hypothesize that the task switch from pure pseudo-colorization
to cell detection is easier and therefore can be learned better with fewer
training samples. The additional masking arguably enhances the learned
representations, leading to better performance in the previous experimental
setups. However, the associated reconstruction of masked image parts differs
more than the pure pseudo-colorization from cell detection using centroid
representations.

###  4.3 Comparing Self-supervised Pre-training Methods on Cell Images

In this set of experiments, we are comparing our proposed pre-training
objective, pseudo-colorizing masked cells (PMC), with recent methods for self-
supervised pre-training on cell images. Following Ciga et al. (2022), we pre-
train our model using the contrastive learning framework SimCLR. In addition,
we compare our method with edge-based self-supervision (EdgeSSV) (Dawoud et
al., 2022) and pure masked image modeling with the masking scheme of masked
autoencoders (MAE) (He et al., 2022). The performance comparison is performed
with backbone evaluation as training procedure. For this, the weights in the
backbone are frozen after the pre-training, so that only the neck part and the
two heads of our model adapt to the downstream task of cell detection.

Experimental setup. As described in Section 4.1, we pre-train with a diverse
set of 6 fluorescence microscopy datasets. Subsequent training to evaluate
performance is performed on the Fluo-C3DH-A549-SIM and the Fluo-N2DH-SIM+
datasets. The pre-training of all methods is performed for 75 epochs, the
subsequent training for 50 epochs. During pre-training and training, we use
the same optimizer setup and learning rate schedule as in Section 4.2 for all
methods. During pre-training, we randomly perform geometric data augmentations
on-the-fly such as flip, rotation, and crop transformations. An exception is
the pre-training of SimCLR, where color affine transformations are performed
in addition to geometric augmentations, as specified in the framework (Chen et
al., 2020). For color affine transformations, the prepended ContextBlock in
our model and the corresponding input tenors with 3 channels additionally
serve as adapters to apply such transformations to grayscale images. For
SimCLR, we restrict the extent of the geometric augmentations, since the
datasets used were created from time-lapse microscopy videos. In these videos,
cells move only moderately between frames, so that successive frames are very
similar. For contrastive learning, this means that strong geometric
augmentations would lead to two differently augmented views of the same frame
being more different from each other than from surrounding frames. However,
since all frames except the two views of the same frame are used as negative
samples in contrastive learning, we limit the extent of geometric
transformations to only slight rotations (+/- 10°) and cropping to a minimum
of 90% of the original size. For SimCLR-pre-training, we replace the two model
heads by one projection head with three fully connected layers, which have
256, 128, and 64 nodes. For the remaining methods, the architecture of our
CellCentroidFromer model is not changed, only the top-layer is switched
according to the output format between pre-training and training. Training
runs are performed three times, we report mean and standard deviation in Table
2.

Table 2: Comparing self-supervised pre-training methods on cell images. Best
scores are bold, second best scores are underlined. A549-SIM and N2DH-SIM+
refer to the Fluo-C3DH-A549-SIM and Fluo-N2DH-SIM+ datasets.

Pre-training datasets
---
N2DH-SIM+ A549-SIM Huh7 MDA231 GOWT1 MSC
Pre-training method | Det. dataset | APtest | AP${}_{test}^{50}$ | AP${}_{test}^{small}$ | AP${}_{test}^{medium}$ | AP${}_{test}^{large}$ | mIoU${}^{heatmap}_{train}$
SimCLR | A549-SIM | 24.40 (2.69) | 75.50 (0.99) | \- (-) | 24.20 (2.12) | 27.10 (3.25) | 71.33 (1.21)
MAE-like | A549-SIM |  57.43 (2.23) | 86.13 (1.75) | \- (-) |  49.83 (2.71) |  71.30 (2.75) |  94.30 (1.08)
EdgeSSV | A549-SIM | 55.70 (3.11) |  87.35 (2.90) | \- (-) | 47.10 (3.25) |  70.40 (2.97) | 93.04 (0.15)
PMC (Ours) | A549-SIM |  57.70 (3.54) |  87.90 (1.41) | \- (-) |  51.80 (2.97) | 70.00 (3.11) |  93.08 (0.80)
SimCLR | N2DH-SIM+ | 24.10 (0.99) | 60.55 (1.06) | 20.10 (0.99) | 37.45 (0.92) | \- (-) | 68.10 (1.72)
MAE-like | N2DH-SIM+ | 27.25 (1.34) | 66.55 (1.63) | 23.40 (1.13) | 39.85 (1.77) | \- (-) | 84.78 (1.44)
EdgeSSV | N2DH-SIM+ |  31.30 (0.71) |  72.50 (1.41) |  26.95 (0.64) |  45.20 (0.57) | \- (-) |  86.23 (1.29)
PMC (Ours) | N2DH-SIM+ |  33.75 (0.64) |  73.10 (0.57) |  28.90 (0.85) |  49.25 (0.64) | \- (-) |  85.15 (1.06)

Results. As expected, all methods perform better on the Fluo-C3DH-A549-SIM
dataset than on the Fluo-N2DH-SIM+ dataset, which contains 3D data but only
one cell is visible per frame. Furthermore, this dataset does not contain
small cells, accordingly we can not compare AP${}_{test}^{small}$ scores. The
cells in this 3D dataset have an ellipsoidal shape, therefore their size in 2D
slices decreases from the center of the volume to the border. Accordingly, the
detection in 2D slices from the center of the volume is evaluated with the
AP${}_{test}^{large}$ metric, and towards the border with the
AP${}_{test}^{medium}$ metric. In the inner 2D slices, cells are detected more
accurately by all methods. On the Fluo-N2DH-SIM+ dataset, medium sized cells
are detected more accurately than small cells by all methods. On both
datasets, SimCLR performs worst overall, which can be explained by the fact
that the similarity between individual frames in time-lapse videos makes
training for SimCLR more challenging. MAE-like masking performs significantly
worse on the Fluo-N2DH-SIM+ dataset than on the Fluo-C3DH-A549-SIM dataset.
This supports our hypothesis that this masking scheme is better suited for
data with one concept (one cell) per image. On the Fluo-C3DH-A549-SIM dataset,
MAE-like masking performs second best and yields APtest scores within 1% of
our proposed method. The EdgeSSV method achieves high AP scores on both
datasets. Overall, our proposed method achieves the highest AP scores on both
datasets. However, considering that on the Fluo-C3DH-A549-SIM dataset the
three best methods yield APtest scores within a range of 2% and the higher
standard deviation values, the performance difference is less significant for
this dataset. Interestingly, our method does not achieve the highest
mIoU${}^{heatmap}_{train}$ scores in either dataset, which is measured on the
training data. This suggests that our method generalizes better and other
methods tend to overfit somewhat more on the training data.

##  5 Discussion

Our experiments reveal that pseudo-colorization is an effective extension to
autoencoding for pre-training on cell images. Pseudo-color autoencoding with
pseudo-spectral colormaps, which approximate the semantics of cell images,
yields the best results on fluorescence microscopy images. As recently shown
for natural images (Assran et al., 2022), the combination of standalone self-
supervision objectives with masked image modeling can also further improve
performance on biomedical cell images. The proposed unique combination of
pseudo-color autoencoding and masked image modeling for cell images can
outperform several autoencoding schemes, contrastive learning, and edge-based
self-supervision. Our proposed masking scheme consists of smaller patches and
covers a much smaller area of images than the typical masking scheme of masked
autoencoders. This prevents object instances from being completely masked. We
found this to be beneficial for subsequent training on object detection in our
experiments. Our CellCentroidFormer model has no special mechanism to suppress
the artificial edges added by masking. Nevertheless, as shown in Figure 1 (c),
no edge artifacts are visible in the reconstructions. We hypothesize that the
self-attention layers in the neck part of our model help to suppress these
edge artifacts, since self-attention can be understood as a form of
generalized spatial smoothing (Park and Kim, 2021). However, it is plausible
that our method can be further improved by using masked convolutions (Gao et
al., 2022) instead of regular convolutions to suppress artificial edges.
Moreover, we do not achieve the computational efficiency of masked
autoencoders where only unmasked patches are processed in the encoder. These
are possible future directions of our work. To this end, we show that our
proposed pre-training method can achieve good performance on small cell image
datasets using an autoencoder-like model without special architectural
modifications.

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Knots from a random matrix theory with replicas 111 This article is a
contribution to the Michael Fisher book for his memory, edited by A. Aharony,
O. Entin-Wohlman, D. Huse and L. Radzihovsky.
Shinobu Hikami

Okinawa Institute of Science and Technology Graduate University,

1919-1 Tancha, Okinawa 904-0495, Japan.

e-mail<EMAIL_ADDRESS>

Abstract

A classical knot is described by a one-stroke trajectory of a string with
entanglements. The replica method is a powerful tool in statistical mechanics
for dealing with string-like objects like polymers or self-avoiding walks. We
consider here the ($N\to 0$) replica limit for Gaussian means of products of
traces of $N\times N$ Hermitian matrices, which correspond to one-stroke
graphs for knots. The Seifert surfaces of knots and links are thus related to
a random matrix model. The zeros of Alexander polynomials on the unit circle
are discussed for the case of n-vertices in analogy with the Yang-Lee edge
singularity. The extension of one-matrix models to higher dimensional knots is
considered, and also to the half-integral level $k$ in a Chern-Simons gauge
theory.

##  1 Introduction

The replica method is a powerful tool in statistical mechanics to deal with
disordered systems. The critical behavior of a single polymer, or equivalently
the geometric properties of self-avoiding walks are given by a zero-replica
limit ($N\to 0$) of the $O(N)$ vector model [1], allowing the use of Wilson-
Fisher $\epsilon$ expansion [2]. The Anderson localization in a random
potential is known to be the zero replica limit of a Grassmannian nonlinear
$\sigma$ model, opening the way to a renormalization group analysis [3]. The
intersection numbers of $p$-spin curves for the moduli space of Riemann
surfaces [13, 5] have also been investigated by a replica method using a
random matrix theory with an external source [6, 7, 8]. The replica limits
($N\to 0$) appears as one stroke trajectories for Wilson lines in a quantum
field theory.

Generally those problems are related to one-stroke trajectories of strings. We
shall examine in this article how the replica method for a Gaussian random
matrix theory corresponds to the theory of knots studied intensively in
mathematics and applied in statistical mechanics (for instance, see the text
book of knots [9]). This classical theory has bee extended to higher
dimensional knots, such as 2-knots [10], which may also be put in
correspondence with matrix models.

##  2 Knot graphs and matrix models

Let us recall first some earlier results on Gaussian matrix integrals. We
consider Gaussian averages as in [8], of the form

$<X(M)>=\frac{1}{Z}\int dMX(M)e^{-\frac{1}{2}{\rm tr}M^{2}}$ (1)

where $X(M)$ is a product of multi-traces of ${\rm tr}M^{n}$, i.e.
$(\prod_{i}{\rm tr}M^{n_{i}})$. The matrices $M$ are Hermitian $N\times N$ and
$Z$ is a normalization, which ensures that $<1>=1$.

For multi-traces such as $<\frac{1}{N}\rm tr(M^{n_{1}}\cdots{\rm
tr}M^{n_{k}})>$ the application of Wick’s theorem produces a sum of Feynman
graphs with double lines between the $k$ vertices. In the $N\to 0$ limit the
only graphs which remain consist of a single stroke line going along the
propagators from vertex to vertex. From our previous work on intersection
numbers of the moduli space of curves on a Riemann surface [6], starting with
the Konsevitch model [13], we have found an explicit formula for the replica
limit of the generating function of multi traces . If

$U(\sigma_{1},...,\sigma_{n})=\frac{1}{N}<\prod_{i=1}^{n}{\rm
tr}e^{\sigma_{i}M}>$ (2)

denotes this generating function, its zero-replica limit is given by

$\lim_{N\to
0}U(\sigma_{1},...,\sigma_{n})=\frac{2^{n}}{\chi^{2}}\prod_{i=1}^{n}{\rm
sinh}\biggl{(}\frac{\chi\sigma_{i}}{2}\biggr{)}$ (3)

with $\chi=\sum_{i=1}^{n}\sigma_{i}$. The expansion of the generating function
(3) in powers of the $\sigma_{i}$ leads easily to explicit results such as
selection rules, for instance,

$\lim_{N\to 0}\frac{1}{N}<({\rm tr}M^{3})^{k}>=0$ (4)

unless $k=2$ ( mod 4). For other powers (1) one finds for instance

$\lim_{N\to 0}\frac{1}{N}<({\rm
tr}M^{3})^{4g-2}>=\frac{3^{3g-2}2^{-2g}(6g-4)!(4g-2)!}{g!(3g-2)!}.$ (5)

In our earlier work $g$ was identified with the genus of the Riemann surface
whose intersection was the r.h.s. of (5). The Gaussian mean of above case in
the replica limit has a number, which counts the different contraction of
legs. For instance, $g=1$ case, this number becomes 3, which is 3
possibilities of contraction of 6 legs of the two vertices in the replica
limit. Thus such numbers are related to the numbers of different knots and
circles.

On the other hand mathematicians describe knots with the help of Seifert
surfaces, whose edge provides a directed knot [9]. A Seifert surface is made
of disks connected each to one another by twisted bands (Seifert band). One
establishes a correspondence between such a surface with a Feynman diagrams,
in which a disk corresponds to a vertex, and a connected Seifert band between
disks as a propagator. From the point of view of knots, this replica limit in
the matrix side involves both knots and unknotted circles : the correspondence
between knots and matrix averages is not one-to-one. The matrix model ignores
the two ways of the over or down trails in the Seifert band. The matrix model
belongs to two dimensions, while knots belong to three dimensions, embedded
into $S^{3}$. The description of directed trails over and down is
characterized by a skein relation [9]. If one endows the vertex of a matrix
graph with a sign, like the head and tail of a coin, the connection of two
vertices (head and tail) produces a Seifert band with the two choices of up or
down crossing.

The standard catalogue of knots (Rolfsen catalogue) lists the knots with their
signature such as $3_{1},4_{1},5_{1},5_{2},...$. The knot of $3_{1}$, named
trefoil, $4_{1}$ is a figure 8 knot etc. The correspondence with matrix
averages in the zero-replica limit is built as follows. For instance, a
trefoil knot ($3_{1}$) corresponds to $\lim_{N\to 0}\frac{1}{N}<{(\rm
tr}M^{3})^{2}>$ in (1) . The trivalent vertex is described by ${\rm tr}M^{3}$.
The average is Gaussian average, but for the Seifert surface, the Seifert band
is taken as a connector (propagator). It is obtained by flipping half of the
vertices, and connect all vertices by Seifert bands.

For the $5_{1}$ knot, the corresponding Gaussian average is $<({\rm
tr}M^{5})^{2}>$. (From now on a factor $1/N$ and a limit $N\to 0$ is meant).
The Gaussian average of this is 165, which is the sum of three different
graphs, and the sum of the numbers of the contractions of these graphs becomes
165. The knot of $5_{1}$ has a contraction number as 5. These different graphs
lead to $5_{1}$ and circles. For the $7_{1}$ knot, the corresponding term is
$<(\rm trM^{7})^{2}>$. In general, the term of $<({\rm tr}M^{n})^{2}>$ gives
the knot, named as $n_{1}$, where $n$ is odd.

The $4_{1}$ (figure-8) knot corresponds to the average $<({\rm tr}M^{4})({\rm
tr}M^{2})^{2}>$. It is easy to find the Gaussian means of the product of
matrices for each Seifert graphs of knots, by the trails which take the
opposite direction path at the crossing points of knot graphs. Gaussian means
of (1) include unknotted circles, therefore we need some selection rule which
separate knots graphs from unknotted circles. The separation of the unknotted
circles is made through a skein relation [9]. The following table lists the
correspondence between the Rolfsen numbers and the Gaussian averages. Table A:
Standard knot notation (left) and Replica limit of Gaussian mean (right) with
Seifert band contractions denoted by $s$.

$\displaystyle 3_{1}\hskip 14.22636pt<({\rm tr}M^{3})^{2}>_{s}$ $\displaystyle
4_{1}\hskip 14.22636pt<({\rm tr}M^{4})({\rm tr}M^{2})^{2}>_{s}$ $\displaystyle
5_{1}\hskip 14.22636pt<({\rm tr}M^{5})^{2}>_{s}$ $\displaystyle 5_{2}\hskip
14.22636pt<({\rm tr}M^{3})^{2}({\rm tr}M^{2})^{2}>_{s}$ $\displaystyle
6_{1}\hskip 14.22636pt<({\rm tr}M^{4})({\rm tr}M^{2})^{4}>_{s}$ $\displaystyle
6_{2}\hskip 14.22636pt<({\rm tr}M^{6})({\rm tr}M^{4})({\rm tr}M^{2})>_{s}$
$\displaystyle 6_{3}\hskip 14.22636pt<({\rm tr}M^{6})({\rm tr}M^{3})^{2}>_{s}$
$\displaystyle 7_{1}\hskip 14.22636pt<({\rm tr}M^{7})^{2}>_{s}$ $\displaystyle
7_{2}\hskip 14.22636pt<({\rm tr}M^{3})^{2}({\rm tr}M^{2})^{4}>_{s}$
$\displaystyle 7_{3}\hskip 14.22636pt<({\rm tr}M^{5})^{2}({\rm
tr}M^{2})^{2}>_{s}$ $\displaystyle 7_{4}\hskip 14.22636pt<({\rm
tr}M^{3})^{2}({\rm tr}M^{2})^{4}>_{s}$ $\displaystyle 7_{5}\hskip
14.22636pt<({\rm tr}M^{5})({\rm tr}M^{4})({\rm tr}M^{3})({\rm tr}M^{2})>_{s}$
$\displaystyle 7_{6}\hskip 14.22636pt<({\rm tr}M^{7})({\rm tr}M^{3})({\rm
tr}M^{2})^{2}>_{s}$ $\displaystyle 7_{7}\hskip 14.22636pt<({\rm tr}M^{6})({\rm
tr}M^{3})^{2}({\rm tr}M^{2})>_{s}$

The same Gaussian means appear for different knots. The difference is due to
the different crossing ways in a Seifert band. For instance, the knots of
$8_{9},8_{17},8_{18},8_{19},8_{20},8_{21}$ have same the same Gaussian means.
They correspond indeed to several different configurations of the crossings in
the Seifert band. The contractions of $8_{9}$, $8_{18}$ and $8_{17}$ are all
different (Feynman diagrams are different). Starting from the alternating knot
of $8_{17}$, by the changes of the paired crossing of over or down of these
Seifert bands, other knots of non-alternating knots $8_{19}$, $8_{20}$ and
$8_{21}$ are obtained.

Some cases appear with the same Gaussian mean but different Seifert graphs.
The cases of $7_{2}$ and $7_{4}$ have the same Gaussian mean $<({\rm
tr}M^{3})^{2}({\rm tr}M^{2})^{4}>_{s}$, but the Seifert graphs are different.

The Gaussian mean is evaluated by Wick contraction, which is a commutative
algebraic procedure. However, if one writes a Feynman diagram in the zero-
replica limit, using for the drawing double line propagators, the order chosen
for writing the diagram matters. According to the order chosen for the double
line propagators, sometimes one generates knotted diagrams, but some ordering
of lines may lead to diferrent knots or even unknotted circles. For instance
in the case of $<({\rm tr}M^{3})^{2}>$, the maximum crossing diagram gives the
trefoil knot $3_{1}$ or a circle, depending on the choices for drawing a
maximum crossing Feyman diagram. This is the same in a Seifert band, which
gives a knot ($3_{1}$) or a circle, depending upon the choices of which line
goes over or down at a crossing.

If we use a standard Feynman diagram, instead a twisted Seifert graphs, there
appear several graphs which are different depending how we write the
propagators in a time order on a paper, i.e. the way of the crossing of the
ribbons. For instance in the case of $<({\rm tr}M^{3})^{2}>$, the maximum
crossing graphs give one-stroke graphs. If the change of order of ribbons
crossing, we have $3_{1}$ and also $4_{1}$. We have mentioned that the head
and tail connected vertex of Seifert surface gives $3_{1}$ or a circle. Thus
there is a difference between a standard Feynman diagram for matrices and
twisted Seifert graphs. We shall discuss the way of the distinctions for
different knots by the numbering in the subsequent section.

##  3 Chern-Simons gauge theory and random matrix

This Seifert band is similar to the propagator of following Chern-Simons
action of gauge field [11].

$\mathcal{L}=\frac{k}{4\pi}\int_{W}{\rm tr}(A\wedge dA+\frac{2}{3}A\wedge
A\wedge A)$ (6)

where $k$ is integer, which corresponds to a level of Lie group, and $W$ is a
three dimensional manifold.

Near the crossing point of Seifert band, two edges are twisted, and it means
the two edges are bound to each other. The two bounded edges are analogous to
two bounded merons, which form an instanton. The merons were discussed in QCD
[16], and they are also found in a super conductor as a vortex, which pairs
condensate into two-dimensional superconductor [17, 18]. These bound states
are described by an abelian or non-abelian gauge theory.

The knot should be related to a Chern-Simons gauge theory, since the knot
invariant polynomial (Jones polynomial) [27] can be found from Chern-Simons
gauge theory with a quantum parameter $q=e^{\frac{2\pi i}{p}}$ with $p=k+2$
[11]. This parameter $p$ has a meaning of a spin $p$, and $p$-spin curve on
the Riemann surface has a topological invariant quantity of an intersection
number of the moduli space [5, 8]

The boundary of three dimensional Chern-Simons, which is turned to two
dimensions, described by Wess-Zumino-Witten (WZW) non-linear sigma model. Thus
matrix model in the replica limit should be connected to WZW or Chern-Simons
gauge theory.

The Seifert surface is interpreted as a matrix model with head and tail two
type vertices. The Seifert band shows the crossing point which has over or
down two choices. This crossing can be interpreted as a twist operator of
Pauli matrices $\sigma_{x}$ and $\sigma_{y}$ which are elements of $su(2)$ Lie
algebra. The vertex of tail can be defined by the vertex multiplied with twist
operator $\sigma$. Then, the standard Wick contraction with replica limit
gives knots. The WZW model for $SU(2)$ Lie group naturally appears by taking
this $su(2)$ Lie algebra with k-valence vertices where $k$ is a level of
$SU(2)_{k}$. The knot polynomial of Jones can be evaluated from the skein
relation with a quantum parameter $q=e^{\frac{2\pi i}{k+2}}$ [11]. The twist
operator, which is coupled to vertices of tail, is similar to the external
source, which is deterministic in a random matrix theory [8].

In this respect, the super matrix $SU(n|m)$ [19] may be useful. The replica
limit is $n\to m$. When $n=2$, this supermatrix becomes $SU(2|2)$ and the
previous Seifert band with the twist operator $\sigma$ will appear.

##  4 Numbering

The correspondence between matrix models, in the zero-replica limit, and knots
can be made one-to-one by numbering the edges. We assign numbers to the edge-
lines, numbering the floor to which this line belongs. The floors are
connected to over or down floors by two types of Seifert bands ( staircase or
escalator). In the case of an over-connection from n-th to (n+1)-th floor, the
number increases of one bit from $n$ to $n+1$. For a downward connection, $n$
becomes $n-1$. At the crossing point of a Seifert band, the total number is
conserved. For incoming numbers $n$ and $m$, the sum of outgoing numbers is
also $n+m$.

For the trefoil knot, the edges of two vertices carrying ${\rm tr}(M^{3})$,
are assigned numberings 1 and 2. The Seifert bands band of this case is $1\to
2$ and $2\to 1$, according to over and down twisted connections. This
assignment is consistent with the fact that the trefoil is an alternating
knot. Along the directed path, the trefoil is written as the sequence of
numbers $(2121212)$, which is expressed as the paired sequence of the
crossings

$\left(\begin{array}[]{cccc}2&1&2&1\\\ 1&2&1&2\end{array}\right)$ (7)

where 3, the sum of the numbers of the first and second row in the same
column, is conserved . This matrix is a sequence of 2 by 2 scattering
matrices. Trivalent vertices can also provide unknotted cases if one Seifert
band can be changed from over to down and vice versa. Then we have

$\left(\begin{array}[]{cccc}2&3&2&1\\\ 1&0&1&2\end{array}\right)$ (8)

Although the number conservation, the sum of incoming and outgoing, holds, the
total sum of the first row is 8, whereas the sum of the second row is 4. It
shows that this case reduces to an unknotted circle.

The number $n$ introduced here may be transformed to a phase $e^{i\theta n}$
with an angle $\theta$. The increase of the number $n$ to $n+1$ leads to a
change $e^{i\theta n}\to e^{i\theta(n+1)}$. This interpretation of phase is
more close to a gauge field picture. This phase is related to parameter of
Jones polynomial $t^{n}=e^{\frac{2i\pi n}{p}}$ if we take $\theta$ as pure
imaginary.

There are many non-alternating knots, which start from the eight crossing
points of Rolsen table as $8_{19},8_{20}$ and $8_{21}$. The knot $8_{17}$ is
an alternating knot. These four knots have same graphs in the replica limit of
matrix model, $<{\rm tr}M^{8}({\rm tr}M^{4})^{2}>_{s}$. The non-alternating
knots have a repeated connections of over or down crossings. Thus the numbers
are increasing or decreasing in several digits. So the numbers are needed to
be greater than 2. For the alternating knot of $8_{17}$, the numbers of
sequence are $(1212121212121212)$, while the non-alternating knot $8_{19}$ has
a sequence $(1232123212323212)$. These two knots have the same graph except
the over or down connections. The following graph represents the knots of
$8_{17},8_{19},8_{20}$ and $8_{21}$. The three strands should be connected as
the upper points to the lowest points. The path has upper to down direction.
Then three directed circles are obtained which are connected by the 8
transverse rungs. The center strand which has 8 rungs represents ${\rm
tr}(M^{8})$, and other both sides column represents ${\rm tr}(M^{4})$. The
rung means the Seifert band, which has two choices of over or down directed
path. For the case $8_{17}$, all segments of the center strand are assigned by
2, and 1 for the vertical segments of strands of both sides. This alternating
knot corresponds to the sequence $(1212121212121212)$. The graph is called
traditionally as ”Amida-kuji” (Kuji means a lottery).

|  |  |
---|---|---|---
|  |  |
|  |  |
|  |  |
|  |  |
|  |  |
|  |  |

This numbering on amida-kuji can also distinguish the unknotted circle from
the linked one. For the link of two circles, the path becomes $2\to 1\to 2$ or
$1\to 2\to 1$. The starting number is same as the ending number. For the
unlinked circles, if they are place in an overlap configuration, the numbers
become $1\to 2\to 3$. The starting number 1 becomes different from the ending
number 3. Thus this case is easily eliminated for the knots and links.

When the average of Gaussian mean is given, and the crossing points are $n$,
it may be not the minimal number of the crossing points. The example is
$<{(\rm tr}M^{4})^{2}{\rm tr}M^{8}>$, which has 8 crossing points. By the
choice of up and down crossing of 8 Seifert bands, one reduces to knot
$5_{2}$, which has 5 crossing points. This example appears in Teneva’s
transformation of a knot $5_{2}$ [12], and it has 4 coloring. The previous
numbering of floor-height becomes a sequence of $12101234323212321$, which
involves a sequence of the increasing numbers $1234$. After Reidemeister
transformation, it reduces to $5_{2}$ knot.

The numbering, which is shown here, is similar to introduction of the time.
The increasing of the height of floor means the increasing of time. The
standard treatment of knot invariants is based on Knitzhnik-Zamolodchikov
equation with the Cauchy repeated integration [13], which uses the integration
of time. We find the one to one correspondence of the replica matrix model by
the numberings to the different knot configurations. This may be same as
Seifert surface, but the replica method has an advantage for generating all
knot configurations.

We consider the numbering of the height of floors. This can be generalized to
any numbers such as a complex number of the root of unity. The general skein
relation is [9]

$xL_{+}-yL_{-}=wL_{0}$ (9)

where $x,y$ and $w$ are arbitrary. One may take these numbers as $x=t^{-1}$,
$y=t$ and $w=(\sqrt{t}-\frac{1}{\sqrt{t}})$, which leads to Jones polynomials
[27].

The skein relation is similar to the algebra of $su(2)$, which has Pauli
matrices $\sigma_{x},\sigma_{y}$ and $\sigma_{z}$ with
$[\sigma_{x},\sigma_{y}]=i\sigma_{z}$. The operator $\sigma_{z}$ corresponds
to eraser of the rung of Amida-kuji, and $\sigma_{x}\sigma_{y}$ and
$\sigma_{y}\sigma_{x}$ correspond to the over and down crossing of Seifert
band. The introduction of the operator $\hat{\sigma}$ for the vertices of the
matrix model is alternative method to the numbering for the knot construction.
This leads to $SU(2)_{k}$ Wess-Zumino-Witten non-linear $\sigma$ model in two
dimensions, which is equivalent Chern-Simon gauge action in three dimensions.

The numbering of the segments of the vertices of the matrix in the replica
limit gives the Chern-Simons gauge field $A$ in (6). The $n$ point function
$U(\sigma_{1},...,\sigma_{n})$ in (2) is similar to the multi-loop invariants
of Chern-Simons theory,

$\hat{\mu}(\sigma_{i},...\sigma_{n})=\int_{A}\prod_{i=1}^{n}{\rm
tr}(e^{\int_{\sigma_{i}}A})d\mu(A)$ (10)

The matrix $M$ with a numbering turns to a gauge invariant field, which has
$SU(2)_{k}$ symmetry. The physical interpretation may be that the one-stroke
trajectory knot is a string (vortex flux ring) which is a three-dimensional
gauge invariant field.

##  5 Derivation of replica formula

For the case of two vertices, we have from (3)

$\displaystyle\lim_{N\to
0}U(\sigma_{1},\sigma_{2})=\sigma_{1}\sigma_{2}+\frac{1}{3}(\sigma_{1}^{5}\sigma_{2}+2\sigma_{1}^{4}\sigma_{2}^{2}+2\sigma_{1}^{3}\sigma_{2}^{3}+\sigma_{1}^{2}\sigma_{2}^{4}+\sigma_{1}\sigma_{2}^{5})+O(\sigma^{10})$
$\displaystyle=\sigma_{1}\sigma_{2}<({\rm
tr}M)^{2}>+\frac{\sigma_{1}^{5}\sigma_{2}}{5!}<{\rm tr}M^{5}{\rm
tr}M>+\frac{\sigma_{1}^{4}\sigma_{2}^{2}}{4!2!}<{\rm tr}M^{4}{\rm tr}M^{2}>$
$\displaystyle+\frac{\sigma_{1}^{3}\sigma_{2}^{3}}{3!3!}<({\rm
tr}M^{3})^{2}>+\frac{\sigma_{1}^{2}\sigma_{2}^{4}}{2!4!}<{\rm tr}M^{4}{\rm
tr}M^{2}>+\frac{\sigma_{1}\sigma_{2}}{5!}<{\rm tr}M{\rm tr}M^{5}>+\cdots$

The coefficient of $\sigma_{1}^{3}\sigma_{2}^{3}$ gives the knot of $3_{1}$,
which is Seifert graph of $<({\rm tr}M^{3})^{2}>$. Other terms are circle
graphs, which are unknotted. From the table A, the knot of two vertices of
$U(\sigma_{1},\sigma_{2})$ are $3_{1},5_{1},7_{1},...$, which are torus knots
and they are represented in the replica limit of the average of product of two
traces $<({\rm tr}M^{2n+1})^{2}>$ (n=0,1,2,…).The number of the trace is the
number of Seifert disks, and it is equal to the number of the vertices of
Seifert graphs.

For three point vertices, the formula of replica limit (3) gives for the order
of $\sigma^{16}$,

$\displaystyle\lim_{N\to
0}U(\sigma_{1},\sigma_{2},\sigma_{3})|_{\sigma^{16}}=\frac{11}{1152}\sigma_{1}^{8}\sigma_{2}^{4}\sigma_{3}^{4}+\frac{11}{1440}\sigma_{1}^{8}\sigma_{2}^{5}\sigma_{3}^{3}+\frac{47}{11520}\sigma_{1}^{8}\sigma_{2}^{6}\sigma_{3}^{2}$
$\displaystyle+\frac{53}{11520}\sigma_{1}^{7}\sigma_{2}^{7}\sigma_{3}^{2}+\frac{347}{34560}\sigma_{1}^{7}\sigma_{2}^{6}\sigma_{3}^{3}+\frac{89}{5760}\sigma_{1}^{7}\sigma_{2}^{5}\sigma_{3}^{4}+\frac{623}{34560}\sigma_{1}^{6}\sigma_{2}^{6}\sigma_{3}^{4}$
$\displaystyle+\frac{511}{23040}\sigma_{1}^{6}\sigma_{2}^{5}\sigma_{3}^{5}+{\rm
symmetric\hskip 5.69054ptterms}$ (12)

which corresponds to the knots of $(8_{8},8_{17},8_{18},8_{19},8_{20},8_{21})$
for $\sigma_{1}^{8}\sigma_{2}^{4}\sigma_{3}^{4}$. The term of
$\sigma_{1}^{8}\sigma_{2}^{5}\sigma_{3}^{3}$ corresponds to
$(8_{16},8_{10},8_{7})$. The term of
$\sigma_{1}^{8}\sigma_{2}^{6}\sigma_{3}^{2}$ corresponds to $(8_{5},8_{2})$
knots.

There is a relation of the knots and the intersection numbers of the curves.
The intersection numbers of the curve is a special case of Gromov-Witten
invariants which maps from the manifold to a point. The Kontsevich Airy matrix
model has an integral representation as [13]

$Z=\int dMe^{\frac{1}{3}{\rm tr}M^{3}+{\rm tr}M^{2}\Lambda}$ (13)

Expanding the term ${\rm tr}M^{3}$, we obtain the power of the vertex ${\rm
tr}M^{3}$ for the Gaussian means as $<({\rm tr}M^{3})^{n}>$ (n is even
number). We found they correspond to $3_{1}$, $9_{23}$, … knots in the
previous section. The numerical values of the Gaussian means are interpreted
as intersection numbers, which are denoted as $<\tau_{n}>$. In the case of
Kontsevich model, the intersection numbers for one marked point are

$<\tau_{1}>=\frac{1}{24},\hskip 14.22636pt<\tau_{3g-2}>=\frac{1}{(24)^{g}g!}$
(14)

The Gaussian means are obtained from the derivatives of the logarithm of the
partition function $Z$ for the matrix model.

The figure eight $4_{1}$ knot has an expression of the Gaussian mean $<({\rm
tr}M^{4})({\rm tr}M^{2})^{2}>$ in table A. This is obtained by the derivatives
of $c_{4}$ and $c_{2}$ for the partition function $Z$, where coefficients of
${\rm tr}M^{j}$ are $c_{j}$,

$Z=\int dMe^{c_{4}{\rm}trM^{4}+c_{2}{\rm tr}M^{2}}$ (15) $<({\rm
tr}M^{4})({\rm
tr}M^{2})^{2}>|_{c_{4}=0,c_{2}=-\frac{1}{2}}=\frac{\partial^{3}}{\partial
c_{4}\partial^{2}c_{2}}{\rm logZ}$ (16)

We consider the k-point function,

$U(\sigma_{1},...,\sigma_{k})=\frac{1}{N}<{\rm tr}e^{\sigma_{1}M}\cdots{\rm
tr}e^{\sigma_{k}M}>$ (17)

where $M$ is $N\times N$ Hermitian random matrix. There is an integral formula
for $k$-point function [6]

$U(\sigma_{1},...,\sigma_{k})=(-1)^{\frac{k(k-1)}{2}}e^{\sum\frac{1}{2}\sigma_{i}^{2}}\oint\prod_{i=1}^{k}\frac{du_{i}}{2i\pi}e^{\sum
u_{i}\sigma_{i}}\prod_{1}^{k}(1+\frac{\sigma_{i}}{u_{i}})^{N}{\rm
det}\frac{1}{u_{i}+\sigma_{i}-u_{j}}$ (18)

In the limit $N\to 0$, we have a logarithmic terms,

$\displaystyle\lim_{N\to
0}\frac{1}{N}U(\sigma_{1},...,\sigma_{k})=(-1)^{\frac{k(k-1)}{2}}e^{\sum\frac{1}{2}\sigma_{i}^{2}}\oint\prod_{i=1}^{k}\frac{du_{i}}{2i\pi}e^{\sum
u_{i}\sigma_{i}}$ $\displaystyle\left(\sum_{i=1}^{k}{\rm
log}(1+\frac{\sigma_{i}}{u_{i}})\right){\rm
det}\frac{1}{u_{i}+\sigma_{i}-u_{j}}$ (19)

When $k=1$ one point function, this integral is evaluated by the cut of
logarithm. The integral of $u$ becomes a line integral $-\sigma<u<0$. (For
$k=1$, the determinant is just $\frac{1}{\sigma}$).

$\displaystyle\lim_{N\to
0}U(\sigma)=\frac{1}{\sigma}e^{\frac{1}{2}\sigma^{2}}\oint\frac{du}{2i\pi}e^{u\sigma}{\rm
log}(1+\frac{\sigma}{u})$
$\displaystyle=\frac{1}{\sigma}e^{\frac{1}{2}\sigma^{2}}\int_{-\sigma}^{0}due^{u\sigma}=\frac{{\rm
sinh}(\frac{\sigma^{2}}{2})}{(\frac{\sigma^{2}}{2})}$ (20)

This result is also obtained by the contour integral for the pole $u=0$ by the
expansion of the logarithmic term,

$\displaystyle U(\sigma)$ $\displaystyle=$
$\displaystyle\frac{1}{\sigma}e^{\frac{\sigma^{2}}{2}}\oint\frac{du}{2i\pi}e^{u\sigma}(\sum_{n=1}^{\infty}(-1)^{n-1}\frac{1}{n}(\frac{\sigma}{u})^{n})$
(21) $\displaystyle=$ $\displaystyle
1+\frac{1}{24}\sigma^{4}+\frac{1}{1920}\sigma^{8}+O(\sigma^{12})$

which leads to (5).

As a categorification of knot theory, Kontsevich loop expansion has been
studied, which provides Vassiliev invariants as coefficients [13, 14]. The
contribution of one loop order is given by ${{\rm
sinh}(\frac{x}{2})}/{(\frac{x}{2})}$, and the logarithm of this one-loop term
becomes a series expansion of $x^{2}$ which coefficients
$B_{n}/(4n\cdot(2n)!)$ where $B_{n}$ is Bernoulli number.

$\sum b_{2n}x^{2n}=\frac{1}{2}{\rm log}(\frac{{\rm
sinh}(\frac{x}{2})}{\frac{x}{2}})$ (22)

This Knotsevich one-loop contribution coincides with (5). Vassiliev invariant
for knot $K$ of order $j$ ( i. e. number of double points in knot $K$) is
denoted by $v_{j}(K)$, and Vassiliev invariants has a generating function
$\sum_{j}v_{j}(K)x^{j}$. This invariants are related to Jones polynomial for
knot $K$, $V_{K}(t)$, with $t=e^{x}$. The expansion of Jones polynomial in
power of $x$ provides Vassiliev invariants $v_{j}(K)$ as [15]

$\sum_{j}v_{j}(K)x^{j}=V_{K}(e^{x})$ (23)

The Vassiliev invariant $v_{2}(K)$ is equal to $(-3)\times$ [$z^{2}$
coefficient in Conway polynomial $P(z)$]. For instance $v_{2}(5_{2})=-6$ and
$P(z)=1+2z^{2}$, where $z=\sqrt{t}-\frac{1}{\sqrt{t}}$. The replica formula of
$U(\sigma_{1},...,\sigma_{n})$, which is used for the intersection theory [8],
is interesting in the connections of Kontsevich loop expansion and Vassiliev
invariants.

For two point function ($k=2$), we have

$\displaystyle\lim_{N\to
0}U(\sigma_{1},\sigma_{2})=-e^{\frac{1}{2}(\sigma_{1}^{2}+\sigma_{2}^{2})}\oint\frac{du_{1}du_{2}}{(2i\pi)^{2}}e^{u_{1}\sigma_{1}+u_{2}\sigma_{2}}$
$\displaystyle\times[({\rm log}(1+\frac{\sigma_{1}}{u_{1}})+({\rm
log}(1+\frac{\sigma_{2}}{u_{2}})]\frac{1}{(u_{1}-u_{2}+\sigma_{1})(u_{2}-u_{1}+\sigma_{2})}$
(24)

If we consider the contour of $u_{1}$, there is no pole for the contour of the
logarithmic. Thus it vanishes. Then the integral becomes the first term,

$\displaystyle\lim_{N\to
0}U(\sigma_{1},\sigma_{2})=-e^{\frac{1}{2}(\sigma_{1}^{2}+\sigma_{2}^{2})}\oint\frac{du_{1}du_{2}}{(2i\pi)^{2}}e^{u_{1}\sigma_{1}+u_{2}\sigma_{2}}$
$\displaystyle\times({\rm
log}(1+\frac{\sigma_{1}}{u_{1}})\frac{1}{(u_{1}-u_{2}+\sigma_{1})(u_{2}-u_{1}+\sigma_{2})}$
(25)

By taking the pole of $u_{2}=u_{1}-\sigma_{2}$ and $u_{2}=u_{1}+\sigma_{1}$,
we have

$\displaystyle
e^{\frac{1}{2}(\sigma_{1}^{2}+\sigma_{2}^{2})}\frac{1}{\chi}(e^{-\sigma_{2}^{2}}-e^{\sigma_{1}\sigma_{2}})\oint\frac{du_{1}}{2i\pi}e^{u_{1}\chi}{\rm
log}(1+\frac{\sigma_{1}}{u_{1}})$ $\displaystyle=\frac{4}{\chi^{2}}{\rm
sinh}\frac{\chi\sigma_{1}}{2}{\rm sinh}\frac{\chi\sigma_{2}}{2}$ (26)

where $\chi=\sigma_{1}+\sigma_{2}$. Repeating this process, we obtain the
replica formula of (3) for $k>2$ [6].

Intersection numbers Kontsevich matrix model is given by

$Z=\int dMe^{\frac{i}{3}{\rm tr}M^{3}-\lambda{\rm tr}M^{2}}$ (27)

and the intersection numbers for the one marked point $<\tau_{n}>_{g}$ is [13]

$<\tau_{n}>_{g}=\sum<\tau_{n}>_{g}t_{n}$ (28)

The values of the intersection numbers for trivalent matrix model (Airy matrix
model) is

$<\tau_{n}>_{g}=\frac{1}{(24)^{g}g!}$ (29)

where $n$ is determined by Riemann-Roch relation of Riemann surface,

$3g-2=n$ (30)

Thus we find the direct relation of trefoil knot to the intersection numbers
of $<\tau_{1}>_{g=1}$. The knots of trivalent Seifert vertices are all
alternating knots, and therefore the Kontsevich Airy matrix model corresponds
to knots $3_{1}$, $9_{23}$,… one to one. Thus the intersection theory of one
marked point gives the knot of trivalent vertices. Links The $N\to 0$ limit of
the matrix expectation values give diagrams that can be traced on a sheet of
paper without lifting a pen : single-stroke diagrams. The next order in $N$
describes two entangled knots, in which the entanglement is due to a double
layout of two continuous intricate lines. There again one can obtain the
number of entangled knots from the same matrix result. For the one point
function, the linking is obtained from the terms of order $N^{2}$ of $N\times
N$ Hermitian matrix.

$\displaystyle
U(\sigma)=\frac{e^{\frac{\sigma^{2}}{2}}}{N\sigma}\oint\frac{du}{2i\pi}e^{\sigma
u}e^{N{\rm log}(1+\frac{\sigma}{u})}$ (31) $\displaystyle=$
$\displaystyle\frac{e^{\frac{\sigma^{2}}{2}}}{\sigma}\oint\frac{du}{2i\pi}e^{\sigma
u}[{\rm log}(1+\frac{\sigma}{u})+\frac{N}{2}[{\rm
log}(1+\frac{\sigma}{u})]^{2}+O(N^{2})$

Thus the one-point one link expectations are obtained from the term of order
$N$ in (31), describing the diagrams made of two strokes trails. The contour
integral in (31) provides

$U(\sigma)={e^{\frac{\sigma^{2}}{2}}}\sum_{1}\frac{(N-1)(N-2)\cdots(N-k+1)}{k!(k-1)!}\sigma^{2k-2}$
(32)

and, in the $N\to 0$-limit

$\displaystyle(N-1)(N-2)\cdots(N-k+1)=(-1)^{k-1}(k-1)!$
$\displaystyle+N(-1)^{k}(k-1)![1+\frac{1}{2}+\cdots+\frac{1}{k-1}]+O(N^{2})$
(33)

Expressing the sum

$1+\frac{1}{2}+\cdots+\frac{1}{k-1}=\int_{0}^{1}dx\frac{1-x^{k-1}}{1-x}$ (34)

one finds the terms of order $N$ in terms of the finite integral [22]

$\displaystyle U(\sigma)|_{N}$
$\displaystyle=\frac{e^{\sigma^{2}/2}}{\sigma^{2}}\int_{0}^{1}\frac{dx}{1-x}[(e^{-\sigma^{2}}-1)-\frac{1}{x}(e^{-x\sigma^{2}}-1)]$
(35)
$\displaystyle=\frac{1}{2}\sigma^{2}+\frac{1}{72}\sigma^{6}+O(\sigma^{8})$

The one point function $U(\sigma)$ for arbitrary $N$ is expressed as

$\displaystyle U(\sigma)$ $\displaystyle=$
$\displaystyle\frac{1}{N\sigma}\oint\frac{du}{2i\pi}e^{\sigma
u}(\frac{1+\frac{\sigma}{2u}}{1-\frac{\sigma}{2u}})^{N}$ (36) $\displaystyle=$
$\displaystyle\frac{1}{\sigma}\sum_{k=1}^{\infty}\frac{N^{k-1}}{k!}\oint\frac{du}{2i\pi}e^{\sigma
u}[{\rm log}(\frac{1+\frac{\sigma}{2u}}{1-\frac{\sigma}{2u}})]^{k}$

For small $\sigma$ expansion becomes

$\displaystyle
U(\sigma)=\sum_{k=1}^{\infty}\frac{N^{k-1}}{k!}[\frac{1}{(k-1)!}\sigma^{2k-2}+\frac{1}{12(k+1)!}\sigma^{2k+2}$
$\displaystyle+(\frac{k}{80}+\frac{k(k-1)}{288})\frac{1}{(k+3)!}\sigma^{2k+6}$
$\displaystyle+(\frac{k}{448}+\frac{k(k-1)}{960}+\frac{k(k-1)(k-2)}{10368})\frac{1}{(k+5)!}\sigma^{2k+10}$
$\displaystyle+(\frac{k}{2304}+\frac{71k(k-1)}{268800}+\frac{k(k-1)(k-2)}{23040}$
$\displaystyle+\frac{k(k-1)(k-2)(k-3)}{497664})\frac{1}{(k+7)!}\sigma^{2k+14}+O(\sigma^{2k+18})]$
(37)

Above expansion is consistent with (5) for $k=1$ and also with (35) for $k=2$.
When the scaling $\sigma^{2}\to\sigma^{2}/N$ is taken, the expansion of the
large $N$ is obtained. The leading order of $N$, it is expressed as

$\lim_{N\to\infty}U(\sigma)=\frac{1}{t}J_{1}(2t)$ (38)

where $\sigma=-it$ and $J_{1}(x)$ is a Bessel function. The next order of
$\frac{1}{N^{2}}$ is also expressed by the Bessel function of $J_{2}$. Thus
the scaling $\sigma\to\sigma/\sqrt{N}$ in (5) makes an interesting relation
which connects the large $N$ and the replica limit $N\to 0$.

For the supermatrices in the previous section, one point function is given
with external source eigenvalues $r_{i}$ and $\rho_{j}$ [19],

$U(\sigma)=\frac{1}{\sigma}\oint\frac{du}{2i\pi}e^{\sigma
u}\prod_{i=1}^{n}(\frac{u-r_{i}+\frac{\sigma}{2}}{u-r_{i}-\frac{\sigma}{2}})\prod_{j=1}^{m}(\frac{u-\rho_{i}-\frac{\sigma}{2}}{u-\rho_{j}+\frac{\sigma}{2}})$
(39)

When the external source $r_{i}$ and $\rho_{j}$ are put to zero, this reduces
to (5) in the limit $n\to m$.

The link graphs are obtained from two point function
$u(\sigma_{1},\sigma_{2})$ in order of $N$. For instance, from $<{\rm
tr}M^{2}{\rm tr}M^{2}>\sigma_{1}^{2}\sigma_{2}^{2}$, with Seifert band
connections, there appear a link and two circles, depending the choice of
crossings. The construction of the Gaussian means are same as knots. There is
a correspondence similar to table A between Gaussian means and links. The
contractions of propagators are given by twisted Seifert bands. For the link
of two knots are described by the diagram of order $N$.

##  6 Characteristic polynomial for trivalent vertices

When Seifert disk (vertex) has trivalent connector, the corresponding Gaussian
mean average becomes $<({\rm tr}M^{3})^{n}>$. The non-vanishing one is the
case of $n=4m-2,m=1,2,3,...$ from (5). The number $m$ is genus $g$ of the
Riemann surface. This provides a series of genus $g$ intersection numbers for
trivalent vertices (Kontsevich model) and a series of knots,
$3_{1},9_{23},...$.

These knot is classified into 2-bridge knot [9], and Seifert matrix
$\tilde{M}$ is made of the diagonal elements plus one line of the same element
1. From the linking of the Seifert bands of this type of knots, it becomes
easy to get Alexander polynomials through the characteristic polynomials,

$\Delta(t)={\rm det}(t\tilde{M}-\tilde{M}^{T})$ (40)

where $\tilde{M}$ is Seifert matrix and $\tilde{M}^{T}$ is a transpose of
$\tilde{M}$. For the knot of the series of the trivalent vertices, Seifert
matrix $\tilde{M}$ has diagonal elements of 2, except the (1,1) element and
(n,n) element, which are 1. For instance, in the case of trefoil $3_{1}$ and
knot of $9_{23}$, Seifert matrix $\tilde{M}$ become

$\tilde{M}_{3_{1}}=\left(\begin{array}[]{cc}1&1\\\ 0&1\end{array}\right)$ (41)
$\tilde{M}_{9_{23}}=\left(\begin{array}[]{cccc}1&1&0&0\\\ 0&2&1&0\\\
0&0&2&1\\\ 0&0&0&1\end{array}\right)$ (42)

Alexander polynomial $\Delta(t)$, which is a characteristic polynomial defined
by (40), has zeros in a complex $t$-plane. In the trivalent vertices of the
previous section, the determinant is similar to Toeplitz determinant, and has
an interesting zero locus.

It is well known that Ising model with a magnetic field has unit circle zero
locus, and there is Yang-Lee edge singularity studied by Fisher [20]. The
critical exponent $\sigma$ for the magnetization $m\sim(h-h_{c})^{\sigma}$ is
$\sigma=(d-2+\eta)/(d+2-\eta)=\frac{\Delta_{\phi}}{d-\Delta_{\phi}}$. The
scale dimension $\Delta_{\phi}=(d-2+\eta)/2$. The exact value of
$\Delta_{\phi}=-\frac{2}{5}$ in Yang-Lee edge singularity is known for two
dimensions. This model is given by the Lagrangian
${\mathcal{L}}=\frac{1}{2}(\partial\phi)^{2}+i(h-h_{c})\phi+ig\phi^{3}$, and
$\epsilon=6-d$ expansion can be obtained. The density of characteristic
polynomial of trivalent knots have a similarity to edge singularity of
$\phi^{3}$ theory.

The locus of the zero of the characteristic polynomial (Alexander polynomial)
is on the arc of the unit circle $x^{2}+y^{2}=1$ in the region
$-\sqrt{\frac{3}{4}}<y<\sqrt{\frac{3}{4}}$ and $\frac{1}{2}<x<1$ as shown in
Fig.1. The density of the zeros increases in the approaching to the edge point
in the complex plane $(x,y)=(\frac{1}{2},\sqrt{\frac{3}{4}})$. This point is
analogous to the edge point of the zeros of Yang-Lee Ising model in a magnetic
field. The distribution of zeros for trivalent vertex knot shows the square
root singularity as the edge singularity. This behavior is originated from the
relation of cusp singularity $y^{2}=x^{3}$ to the trefoil knot. This is also
related to Landau- Ginzburg potential for primary fields, which are obtained
from matrix models [5, 21].

The knot $5_{1}$ has a vertex with 5 external legs. This is related to
$p$-spin curve with $p=4$. In this case, the same ladder structure for higher
crossing knots made of vertices with 5-legs only as expressed as $<(\rm
trM^{5})^{n}>$, $n=3,6,...$. The zeros of Alexander polynomials are located on
the unit circle similar to the trivalent vertices. The region of the zeros on
the unit circle is limited to the region $(-1)^{\frac{3}{5}}<x$, where $x$ is
the real part of the zeros. The zeros at $(-1)^{\frac{1}{5}}$ and
$(-1)^{\frac{3}{5}}$, for positive imaginary region, appear independent of
$n$, and they are the end points of edges of a spectrum.

Figure 1: Zeros of Alexander polynomial for trivalent vertex, which are on an
arc of the unit circle. The edge points of spectrum are at
$(x,y)=(\frac{1}{2},\pm\sqrt{\frac{3}{4}})$ in a complex coordinate.

##  7 Higher dimensional knots and singularities

The classical knot theory concerns with the embedding of $S^{1}$ into $S^{3}$.
The extension of this knot to higher dimensional case has been studied with
covering spaces of the classical knots [9, 10]. $n$-dimensional knot $(N,M,k)$
is embedding of $n$-dimensional manifold $N^{n}$ in $(n+2)$ dimensional
manifold $M^{n+2}$, ($k:N^{n}\subset M^{n+2}$). The embedding
$k$:$N^{n}\subset M^{m}$ with codimension $m-n\geq 3$ leads to only
unknotting. So we need the codimension 2 for knots. A classical knot is
1-knot, $n=1$. By spinning classical knots, the surface knot (2-knots) is
obtained, which is embedded in 4-manifold ($S^{2}\subset S^{4}$)).

For 2-knot, the Seifert surface is constructed by the spinning of a knot
around a certain axis, which leads to generation of surfaces [23]. Natural way
of 2-knot is obtained from 1-knot by taking the movement in a time direction,
i.e. movie picture [10].

The replica limit of (3) is applied for the time dependent Gaussian matrix
model [21]. By adding ${\rm tr}(dM/dt)^{2}$ term to action, the matrix
describes the movement on time. This time dependent matrix model reduces to
two matrix model by a path integral formulation [21].

When 1-knot is embedded in to 4 dimensions, the knot becomes resolved due to
the additional dimension of time. However, the pair of strings, which makes a
surface becomes knotted. The axis of the spun knot correspond the interaction
of two matrices $M_{1}$ and $M_{2}$ coupled as $(c{\rm tr}M_{1}M_{2})^{2}$.

The 2-knot has singularities such as double point and triple point, which are
generated by the degeneracies of Reidemeister relation in the moving picture
[10]. The Reidemeister relations for 1-knot are generalized to Roseman
relations. Over and down distinction in 1-knot with previous numberings does
not make a sense for 2-knot. The coloring (numbering) may be useful instead
[12]. We consider the replica limit of the time dependent matrix model for the
description of the surface knot.

The two point correlation function $U(\sigma_{1},\sigma_{2})$ is

$U(\sigma_{1},\sigma_{2})=\frac{1}{Z}\int dM_{1}dM_{2}({\rm
tr}e^{\sigma_{1}M_{1}}{\rm tr}e^{\sigma_{2}M_{2}})e^{-\frac{1}{2}{\rm
tr}(M_{1}^{2}+M_{2}^{2}-2cM_{1}M_{2})}$ (43)

The parameter $c$ is a coupling constant of two matrices, which is equal to
$e^{-t}$ with a time $t$. In the limit $t\to\infty$, $c$ becomes vanishing,
and we obtain two non-interacting matrices.

For the picture of the spinning, we take order the terms of $c^{2}$, which
represent the axis of the spinning. The n-point correlation function for
$c\neq 0$ can be formulated in a contour integrals [21]. The replica limit for
this case is obtained by taking $N\to 0$ in the contour integral. For
instance, the two point case is expressed as

$\displaystyle U(\sigma_{1},\sigma_{2})$ $\displaystyle=$
$\displaystyle-\frac{1}{N}\oint\frac{dudv}{(2i\pi)^{2}}(1+\frac{\sigma_{1}}{u})^{N}(1+\frac{\sigma_{2}}{cv})^{N}$
(44) $\displaystyle\times$
$\displaystyle\frac{1}{(u-v+\frac{\sigma_{1}}{N})(u-v+\frac{\sigma_{2}}{N})}e^{\frac{\sigma_{1}u}{1-c^{2}}+\frac{\sigma_{2}v}{1-c^{2}}-\frac{\sigma_{1}^{2}}{2N(1-c^{2})}+\frac{\sigma_{2}^{2}}{2N(1-c^{2})}}$

##  8 Extension to half integer spins

Recently the extension of the $p$ spin curve of integral value of $p$ to half
integer $p$ has been considered by Brézin and myself [24, 25, 26]. The skein
relation is expressed with a parameter $z=\sqrt{t}-\frac{1}{\sqrt{t}}$. This
parameter $t$ appears in Alexander polynomials and Jones polynomials [27].
This parameter $t$ can be a root of unity, $t=e^{\frac{2\pi i}{p}}$. When $p$
is half-integer, the skein relation provides a new relation of knots. There
appears also a factor $\tilde{z}=\sqrt{t}+\frac{1}{\sqrt{t}}$, which
represents a trivial link . It is interesting to note that this factor appears
in the contour integral with a change variable from $u$ to $y$ as
$u=\frac{i}{2}(y^{2}-\frac{1}{y^{2}})$, which gives a Laurent series in $y$
[24, 25, 26]. In the expression of the integrand for correlation function,
there appears a factor $\tilde{z}=\sqrt{t}+\frac{1}{\sqrt{t}}$ with
$y=\sqrt{t}$. The $\mu$ component trivial link is expressed as
$V_{O_{\mu}}(t)=(-1)^{\mu-1}(\sqrt{t}+\frac{1}{\sqrt{t}})^{\mu-1}$, which
appears in the construction of Jones polynomials [9].

This half-integer spin leads to Fermionic case. The case $p=\frac{1}{2}$ is
Dirac spin and $p=\frac{3}{2}$ is Rarita-Schwinger case. Such half integer $p$
and half integer level $k$, ($p=k+2$) has a new properties for knots. Since
$k$ is a coefficient of Chern-Simons action, this fractional level leads to a
fractional charge and a new conformal field theory, which may have application
on topological semi-metals of spin $\frac{3}{2}$ such as Half-Heusler alloy
(PdBiSe etc.).

##  9 Discussion

In this article, we point out explicitly that the expectation values of
vertices for the Gaussian Hermitian matrix model, corresponds to knots in the
zero-replica limit. The diagrammatic expansions show an explicit
correspondence between Seifert graphs and Gaussian means after introduction of
an height function. By the tuning of an external matrix source, the Hermitian
matrix model can be chosen to generate $p$-spin curves [8]. We found the
correspondence of the $n$-vertex Gaussian means to the singularity of Landau-
Ginzburg potentials. This correspondence appears in the distribution of zeros
of the corresponding Alexander polynomial, which is related to Yang-Lee edge
singularity.

The replica limit of $O(N)$ vector model has been used for polymers as a way
to implement the self-avoiding behavior in $d$ dimensions, where $1<d<4$.
Recently, polymers have been studied by a conformal bootstrap method for
general space dimensions $d$ [28, 29]. The Yang-Lee edge singularity has been
studied by the conformal bootstrap in a determinant method [30, 31] for
dimensions $2\leq d\leq 6$. The value $\Delta_{\phi}\sim-\frac{2}{5}$ has been
found for two dimensions. The $d$ dimensional branched polymer is equivalent
to Yang-Lee edge singularity with the dimensional reduction $d\to d-2$ , which
can be seen explicitly in Wilson-Fisher $\epsilon$ expansion [2, 29]. It is
interesting to investigate the dimensional reduction [32] for the knots of
higher dimensions. This will be an interesting future work.

Recently, in a four-dimensional ${\mathcal{N}}=4$ super Yang-Mills theory,
similar replica formula of (3) appears (see footnote 15 in ref. [33]). This
may be an indication that a gauge theory appears from the random matrix theory
with a replica.

The spin glass in a disordered system has been discussed by a replica method
as a replica symmetry breaking [34], which may be equivalent to $p$-adic
theory [35]. We have seen the knot or unknotted circle appear in the random
matrix theory with replica limit. The unknotted circles provide a large
entropy, and a glass transition may be considered as a transition from the
knotted system to unknotted (circle) transitions. It may be interesting to
apply the present study of the replica limit of the matrix model to such glass
transition and gauge glass transition as a vortex lattice melting in a
superconductor [36, 37].

Acknowledgement Author thanks Edouard Brézin for many discussions about
replica method and fractional $p$ spin curves. He thanks Andreani Petrou for
the help of evaluation of knots. He is supported by JSPS KAKENHI 19H01813.

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