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# Lung infection and normal region segmentation from
CT volumes of COVID-19 cases

Masahiro ODA  Information Strategy Office, Information and Communications,
Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi, 464-8601, Japan
Graduate School of Informatics, Nagoya University, Nagoya, Japan  Yuichiro
HAYASHI  Graduate School of Informatics, Nagoya University, Nagoya, Japan
Yoshito OTAKE  Graduate School of Science and Technology, Nara Institute of
Science and Technology, Nara, Japan  Research Center for Medical Bigdata,
National Institute of Informatics, Tokyo, Japan  Masahiro HASHIMOTO
Department of Radiology, Keio University School of Medicine, Tokyo, Japan
Toshiaki AKASHI  Department of Radiology, Juntendo University, Tokyo, Japan
Kensaku MORI  Information Strategy Office, Information and Communications,
Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi, 464-8601, Japan
Graduate School of Informatics, Nagoya University, Nagoya, Japan  Research
Center for Medical Bigdata, National Institute of Informatics, Tokyo, Japan

###### Abstract

This paper proposes an automated segmentation method of infection and normal
regions in the lung from CT volumes of COVID-19 patients. From December 2019,
novel coronavirus disease 2019 (COVID-19) spreads over the world and giving
significant impacts to our economic activities and daily lives. To diagnose
the large number of infected patients, diagnosis assistance by computers is
needed. Chest CT is effective for diagnosis of viral pneumonia including
COVID-19. A quantitative analysis method of condition of the lung from CT
volumes by computers is required for diagnosis assistance of COVID-19. This
paper proposes an automated segmentation method of infection and normal
regions in the lung from CT volumes using a COVID-19 segmentation fully
convolutional network (FCN). In diagnosis of lung diseases including COVID-19,
analysis of conditions of normal and infection regions in the lung is
important. Our method recognizes and segments lung normal and infection
regions in CT volumes. To segment infection regions that have various shapes
and sizes, we introduced dense pooling connections and dilated convolutions in
our FCN. We applied the proposed method to CT volumes of COVID-19 cases. From
mild to severe cases of COVID-19, the proposed method correctly segmented
normal and infection regions in the lung. Dice scores of normal and infection
regions were 0.911 and 0.753, respectively.

###### keywords:

COVID-19, lung, segmentation, fully convolutional network

##  1 INTRODUCTION

Novel coronavirus disease 2019 (COVID-19) was recognized in December 2019. It
spreads over the world causing the large number of infected patients. The
total numbers of cases and deaths related to COVID-19 are more than 103
million and 2 million in the world by February 1, 2021 [1]. Providing
appropriate treatments to a patient and prevention of infection based on
diagnosis result of the patient are important. However, because of the rapid
increase of COVID-19 patients, medical institutions are suffering from a
manpower shortage. Development of a computer aided diagnosis (CAD) system for
COVID-19 is pressing demanded to reduce load on medical staffs. Reverse
transcriptase polymerase chain reaction testing (RT-PCR) is used to diagnose
COVID-19. However, the sensitivity of RT-PCR is not high, ranging from 42% to
71% [2]. In contrast, the sensitivity of chest CT image-based COVID-19
diagnosis is reported as 97% [3]. Chest CT is effective for diagnosis of viral
pneumonia including COVID-19. CT image-base CAD systems are important in
COVID-19 diagnosis. To develop such CAD systems, development of a quantitative
analysis method of condition of the lung is required. Ground-glass opacities
(GGOs) and consolidations are commonly found in the lung regions of viral
pneumonia cases. We call regions including GGO and consolidation as infection
region. The remaining region in the lung is defined as a normal region.
Quantitative analysis of such regions is useful in lung condition analysis.

Previously, some segmentation methods of infection and normal regions from CT
volumes of COVID-19 cases were proposed [4, 5]. Inf-Net [4] segments infection
regions from CT volumes. The method uses reverse attention and edge attention
in their FCN-based segmentation model. It obtained good segmentation accuracy
even from small dataset that contains 45 cases of CT volumes. Yan et al. [5]
proposed a segmentation method of infection and normal regions. They used
progressive atrous spatial pyramid pooling in their FCN model. They trained
the model using large dataset that contains 731 cases of CT volumes.

We propose an automated segmentation method of infection and normal regions in
the lung from CT volumes using a fully convolutional network (FCN) for
COVID-19 segmentation. In the segmentation process, both local and global-
level features are important to segment infection regions that have various
sizes and shapes. We use an COVID-19 segmentation FCN that has dense pooling
connections and dilated convolutions to segment infection regions accurately.
The dense pooling connections enables utilization of multi-scale spatial
features that are extracted in the encoder of FCN in the bottleneck layer. The
dilated convolutions also contribute to extract multi-scale spatial features.
These components in the FCN improve segmentation performances of target
structures that have large variations in their sizes and shapes. Also, these
components contribute to achieve high generalization ability on the FCN even
from small dataset. It was confirmed in our evaluation using a small dataset.

The contributions of this paper can be summarized as: (1) proposal of the
COVID-19 segmentation FCN that has high performance in segmentation of
infection regions which have large variations in their sizes and shapes and
(2) proposal of the FCN that can achieve high generalization ability from
small dataset.

##  2 METHOD

###  2.1 Overview

One application of the proposed method is CAD system for COVID-19 case. CT
volumes of various slice thickness (1.0 to 5.0 mm) are used in diagnosis. The
proposed method is designed to accept various slice thickness by employing a
multiple axial slice processing approach.

A CT volume containing the lung region is processed by the proposed method.
The COVID-19 segmentation FCN performs segmentation from multiple axial
slices. Sequence of axial slices obtained from the CT volume is given to the
FCN. The FCN segments infection and normal regions in the lung. The slice-wise
segmentation results are reconstructed as an output volume.

###  2.2 Image preparation

Input of the FCN is a sequence of axial slices obtained from the input CT
volume. All axial slices are scaled to the image size of 384$\times$384
pixels. CT values in the range from -2050 to 950 H.U. are normalized to the
range from -1.0 to 1.0. Three axial slices of consecutive slice numbers are
combined in a three color channel image. The three color channel image is used
as the input of the FCN.

Ground truth images of the region contain three regions including normal
region, infection region, and air region outside the body.

###  2.3 COVID-19 segmentation FCN

U-Net [6] is commonly used in image segmentation. While the U-Net is effective
for segmentation of convex objects such as the lung and liver, small or thin
objects are easily missed. This is because features of small structures are
lost in multiple max-pooling operations in the encoding path in the U-Net.
Infection regions have various shapes including small, thin, and complicated
shapes. Infection regions are difficult to segment by using U-Net. Recent work
trying to improve segmentation accuracy of such regions. Dense pooling
connection [7] was proposed to utilize spatial information of multiple scales
in layers in the encoding path and the bottleneck layer to perform
segmentation. The bottleneck layer in the U-Net performs feature conversions
from input to output images. However, spatial information in the input image
is lost in max pooling operations in the encoding path. The dense pooling
connection provides spatial information to the bottleneck layer. Furthermore,
mixed poolings [7] are used in the dense pooling connections instead of max
poolings to reduce information loss by max pooling operations. The mixed
pooling is implemented as a combination of max and average poolings. Dilated
convolution [8] was proposed to utilize sparsely-distributed features in
convolution operations. It performs a sparse convolution operation from
feature maps. By combining results of dilated convolution operations of
multiple dilation rates, we obtain a convolution result of multiple-scales in
a feature map. We utilize these techniques in our segmentation.

The structure of our COVID-19 segmentation FCN is shown in Fig. 1. We employ
the dense pooling connections in the encoding path of the FCN. Also, we
introduce two dilated convolution blocks in the encoding path. The dilated
convolution block consists of parallelly-connected dilated convolutions of
multiple dilation rates. The results of dilated convolutions are combined
using a concatenate operation.

Figure 1: COVID-19 segmentation FCN. White boxes are feature maps or images.
Numbers below boxes are numbers of kernel or color channel. Dense pooling
connections are represented as red connections, which are implemented as
combination of mixed poolings. Two dilated convolution blocks are inserted to
encoding path. Structure of dilated convolution block is shown in box.

###  2.4 Segmentation

The FCN is trained using CT volumes for training and corresponding ground
truth volumes. In the inference step, images obtained from CT volumes for
inference are given to the trained FCN. Segmentation results of normal and
infection regions are reconstructed as the output volume.

##  3 EXPERIMENTS AND RESULTS

We applied the proposed method to 20 CT volumes of COVID-19 patients. The CT
volumes were taken in multiple medical institutions in Japan. Specifications
of the CT volumes are: image size was 512$\times$512 pixels, slice number was
56 to 722, pixel spacing was 0.63 to 0.78 mm, and slice thickness was 1.00 to
5.00 mm. Ground truth volumes were checked by a radiologist. In the training
of the FCN, the number of minibatch was 10 and the training epoch number was
100. The generalized dice loss [9] was used as a loss function.

We performed a 10-fold cross validation in our evaluation. The evaluation
criterion is the dice score of normal and infection regions.

Segmentation results are shown in Fig. 2. The proposed method achieved 0.911
and 0.753 dice scores of normal and infection regions, respectively. As a
comparison, we performed segmentation using U-Net [6] and obtained 0.898 and
0.741 dice scores of normal and infection regions. The dice scores were
compared with the previously proposed methods [4, 5] and summarized in Table
1. Among these methods, the proposed method achieved the highest dice score of
infection region.

Table 1: Dice scores of proposed and previous segmentation methods for COVID-19 cases. The highest score among each region is indicated by bold text. | Dice score
---|---
Method | Normal region | Infection region
Inf-Net[4] | - | 0.739
Yan et al.[5] | 0.987 | 0.726
U-Net[6] | 0.898 | 0.741
Proposed method | 0.911 | 0.753
---
(a)
(b)
(c)
Figure 2: Results from (a) chest CT of 1.25 mm slice thickness, (b) CT volume
of 1.00 mm slice thickness including chest and abdominal regions, and (c)
chest CT of 5.00 mm slice thickness. Green and red regions are normal and
infection regions. Top and bottom columns show axial and coronal slices.

##  4 DISCUSSION

The proposed method has a high performance in segmentation of infected region
from CT volumes of COVID-19 patients. It is shown in the Table 1. The result
proofs the proposed method has high generalization ability because the method
obtained the highest dice score even from training on the small dataset. The
method proposed by Yan et al. [5] achieved the highest dice score of normal
region. Because they used a large dataset that contains 731 CT volumes, their
FCN got good segmentation performance. The proposed method is expected to
achieve higher dice scores than results shown in the Table 1 if it is trained
on a large dataset. We will continue working to obtain more data to train the
proposed method.

CT volumes contain not only chest but also abdominal regions as shown in Fig.
2 (b). The proposed method successfully segmented lung regions regardless of
scan range of CT volumes. It indicates our 2D axial slice-based process has
enough performance to segment 3D lung regions.

The slice thickness of CT volumes varies from 1.00 to 5.00 mm. Figures 2 (b)
and (c) show segmentation results from CT volumes of 1.00 and 5.00 mm slice
thicknesses. Even though there is a difference of resolution along the body
axis, the proposed method segmented both normal and infection regions. This is
a benefit of employing 2D slice-based process. 3D image-based process is
difficult to perform segmentation from CT volumes having inhomogeneous
resolutions.

Variation of the shape and intensity of infection regions makes segmentation
difficult. Because we included GGO and consolidation in the infection region,
intensities, distribution pattern has large variation. Therefore, the dice
score of infection regions were lower than normal region. We will improve the
segmentation accuracy of infection regions by adding more training data and
introducing data augmentation techniques.

##  5 CONCLUSIONS

We proposed an automated segmentation method of infection and normal regions
in the lung from CT volumes. The method utilizes a COVID-19 segmentation FCN
that has the dense pooling connections and pararelly-connected dilated
convolutions to segment infection regions that have various shapes and
intensities. In the evaluation utilizing 20 CT volumes of COVID-19 patients,
we achieved segmentation accuracies of 0.911 and 0.753 in dice scores from
normal and infection regions in the lung.

###### Acknowledgements.

Parts of this research were supported by the AMED Grant Numbers
18lk1010028s0401, JP19lk1010036, and JP20lk1010036, the MEXT/JSPS KAKENHI
Grant Numbers 26108006, 17H00867, and 17K20099, the JSPS Bilateral
International Collaboration Grants. We used the Japan Medical Image Database
(J-MID) created by the Japan Radiological Society with support by the AMED
Grant Number JP20lk1010025.

## References

  * [1] “Coronavirus Update.” https://www.worldometers.info/coronavirus/.  (Accessed: February 1, 2021).
  * [2] Simpson, S., Kay, F. U., Abbara, S., Bhalla, S., Chung, J. H., Chung, M., Henry, T. S., Kanne, J. P., Kligerman, S., Ko, J. P., and Litt, H., “Radiological society of north america expert consensus statement on reporting chest CT findings related to COVID-19. endorsed by the society of thoracic radiology, the american college of radiology, and rsna,” Radiology: Cardiothoracic Imaging 2(2) (2020).
  * [3] Ai, T., Yang, Z., Hou, H., Zhan, C., Chen, C., Lv, W., Tao, Q., Sun, Z., and Xia, L., “Correlation of chest CT and RT-PCR testing for coronavirus disease 2019 (covid-19) in china: A report of 1014 cases,” Radiology 296(2) (2020).
  * [4] Fan, D.-P., Zhou, T., Ji, G.-P., Zhou, Y., Chen, G., Fu, H., Shen, J., and Shao, L., “Inf-Net: Automatic COVID-19 lung infection segmentation from CT images,” IEEE Transactions on Medical Imaging 39(8), 2626–2637 (2020).
  * [5] Yan, Q., Wang, B., Gong, D., Luo, C., Zhao, W., Shen, J., Shi, Q., Jin, S., Zhang, L., and You, Z., “COVID-19 chest CT image segmentation - a deep convolutional neural network solution,” arXiv:2004.10987 (2020).
  * [6] Ronneberger, O., Fischer, P., and Brox, T., “U-Net: Convolutional networks for biomedical image segmentation,” International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI) 9351, 234–241 (2015).
  * [7] Playout, C., Duval, R., and Cheriet, F., “A multitask learning architecture for simultaneous segmentation of bright and red lesions in fundus images,” International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI) 11071, 101–108 (2018).
  * [8] Yu, F. and Koltun, V., “Multi-scale context aggregation by dilated convolutions,” International Conference on Learning Representations (ICLR) (2016).
  * [9] Sudre, C. H., Li, W., Vercauteren, T., Ourselin, S., and Cardoso, M. J., “Generalised dice overlap as a deep learning loss function for highly unbalanced segmentations,” International Workshop on Deep Learning in Medical Image Analysis (DLMIA) 10553, 240–248 (2017).


classes $(1,1)$, $(0,1)$, $(1,0)$ in $H_{1}(S_{1,1})\cong\mathbb{Z}^{2}$, and
define an ideal triangulation
$\tau_{k}=\varphi_{1}\circ\varphi_{2}\circ\dots\circ\varphi_{k}(\tau_{0})$ for
each $k=1$, $2$, …, $k_{0}$. Then, the character
$[\bar{r}]\in\mathcal{X}_{\mathrm{PSL}_{2}(\mathbb{C})}(S)$ is associated to
an edge weight system $(a_{k},b_{k},c_{k})$ for the edges $e_{k}$, $f_{k}$,
$g_{k}$ of $\tau_{k}$ that satisfies the induction relation that

$\displaystyle\qquad\qquad a_{k}$ $\displaystyle=b_{k-1}^{-1}$ $\displaystyle
b_{k}$ $\displaystyle=(1+b_{k-1})^{2}a_{k-1}$ $\displaystyle c_{k}$
$\displaystyle=(1+b_{k-1})^{-2}b_{k-1}^{2}c_{k-1}$

if $\varphi_{k}=L$, and

$\displaystyle\qquad\qquad a_{k}$ $\displaystyle=c_{k-1}^{-1}$ $\displaystyle
b_{k}$ $\displaystyle=(1+c_{k-1})^{2}b_{k-1}$ $\displaystyle c_{k}$
$\displaystyle=(1+c_{k-1})^{-2}c_{k-1}^{2}a_{k-1}$

if $\varphi_{k}=R$.

  3. 3.

The first and last edge weight systems $(a_{0},b_{0},c_{0})$ and
$(a_{k_{0}},b_{k_{0}},c_{k_{0}})$ are equal.

###### Proof.

We can simplify the ideal triangulation sweep $\tau=\tau^{(0)}$, $\tau^{(1)}$,
…, $\tau^{(k_{0}-1)}$, $\tau^{(k_{0})}=\varphi(\tau)$ by arranging that, if
$\tau^{(k)}$ differs from $\tau^{(l)}$ by an edge relabelling, then
necessarily $l=k$ or $k\pm 1$. Indeed, the compatibility condition between
edge weight systems corresponds to the case $q=1$ of the Chekhov-Fock
coordinate change isomorphisms of §3.3, and in particular satisfy the relation
$\Phi^{1}_{\tau^{\prime}\tau^{\prime\prime}}\circ\Phi^{1}_{\tau^{\prime\prime}\tau^{\prime\prime\prime}}=\Phi^{1}_{\tau^{\prime}\tau^{\prime\prime\prime}}$.
It follows that, if $\tau^{(k)}$ differs from $\tau^{(l)}$ by a permutation of
its edges, the edge weight system $a^{(k)}$ and $a^{(l)}$ coincide up to the
same permutation. If $|l-k|>1$, we can then shorten the ideal triangulation
sweep to progressively eliminate this situation.

We can similarly arrange that no $\tau^{(k)}$ is equal to an edge relabelling
of $\varphi(\tau^{(l)})$ for some $l>0$.

We then temporarily omit the edge labellings in this ideal triangulation
sweep, and obtain a sequence of unlabelled ideal triangulations
$\tau=\tau_{0}$, $\tau_{1}$, …, $\tau_{k_{0}-1}$, $\tau_{k_{0}}=\varphi(\tau)$
such that each $\tau_{k}$ is obtained from $\tau_{k-1}$ by a diagonal
exchange. By construction, each $\tau_{k}$ corresponds to a single
triangulation $\tau^{(l)}$, or to two consecutive $\tau^{(l)}$ and
$\tau^{(l+1)}$ differing by an edge relabelling, in the sweep.

For every $k\geqslant 1$, we now label the edges of $\tau_{k}$ as $e_{k}$,
$f_{k}$, $g_{k}$ in such a way that $e_{k}$ is the edge that is not an edge of
$\tau_{k-1}$, namely is the new diagonal is the diagonal exchange from
$\tau_{k-1}$ to $\tau_{k}$; the edges $e_{k}$, $f_{k}$ and $g_{k}$ are then
uniquely determined by the property that they occur clockwise in this order
around the faces to $\tau_{k}$. For $k=0$, we set
$e_{0}=\varphi^{-1}(e_{k_{0}})$, $f_{0}=\varphi^{-1}(f_{k_{0}})$,
$g_{0}=\varphi^{-1}(g_{k_{0}})$.

By construction, each ideal triangulation $\tau_{k}$ is equal, up to edge
relabelling, to a triangulation $\tau^{(k^{\prime})}$ of the original ideal
triangulation sweep. The edge weight system $a^{(k^{\prime})}$ of this ideal
triangulation sweep then gives an edge weight system $(a_{k},b_{k},c_{k})$ for
$\tau_{k}$, where the weights $a_{k}$, $b_{k}$ and $c_{k}\in\mathbb{C}^{*}$
are respectively associated to the edges $e_{k}$, $f_{k}$, $g_{k}$ of
$\tau_{k}$. Since $\tau_{k}$ is obtained from $\tau_{k-1}$ by a diagonal
exchange followed with an edge relabelling, each $(a_{k},b_{k},c_{k})$ is
obtained from $(a_{k-1},b_{k-1},c_{k-1})$ by formulas similar to those we
encountered in §3.2, except that we are now in the case of the one-puncture
torus. These precise formulas can, for instance, be found in [Liu09, §2] or
[BL07, §8]; see also below.

We now choose the identification $H_{1}(S_{1,1})\cong\mathbb{Z}^{2}$ so that
the edges $e_{0}$, $f_{0}$, $g_{0}$ are respectively disjoint from closed
curves representing the homology classes $(1,1)$, $(0,1)$, $(1,0)$.

$f_{k-1}$ $e_{k-1}$ $e_{k-1}$ $g_{k-1}$ $g_{k-1}$ $e_{k}$ $f_{k}$ $f_{k}$
$g_{k}$ $g_{k}$ and $\varphi_{k}=L$

$g_{k-1}$ $f_{k-1}$ $f_{k-1}$ $e_{k-1}$ $e_{k-1}$ $e_{k}$ $f_{k}$ $f_{k}$
$g_{k}$ $g_{k}$ and $\varphi_{k}=R$

Figure 5. The two types of diagonal exchanges

For each $k\geqslant 1$, the fact that $\tau_{k}$ is different from
$\tau_{k-2}$ (and from $\varphi^{-1}(\tau_{k_{0}-1})$ for $k=1$) shows that
$e_{k}\neq e_{k-1}$. Therefore, there are only two possibilities:

  1. 1.

the diagonal exchange occurs along the edge $f_{k-1}$, in which case
$f_{k}=e_{k-1}$ and $g_{k}=g_{k-1}$ and the edge weight systems
$(a_{k},b_{k},c_{k})$ and $(a_{k-1},b_{k-1},c_{k-1})$ are related by the
formulas

$\displaystyle\qquad\qquad a_{k}$ $\displaystyle=b_{k-1}^{-1}$ $\displaystyle
b_{k}$ $\displaystyle=(1+b_{k-1})^{2}a_{k-1}$ $\displaystyle c_{k}$
$\displaystyle=(1+b_{k-1})^{-2}b_{k-1}^{2}c_{k-1};$

in this case we set $\varphi_{k}=L=\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)$;

  2. 2.

or the diagonal exchange occurs along the edge $g_{k-1}$, in which case
$f_{k}=f_{k-1}$, $g_{k}=e_{k-1}$ and the relations

$\displaystyle\qquad\qquad a_{k}$ $\displaystyle=c_{k-1}^{-1}$ $\displaystyle
b_{k}$ $\displaystyle=(1+c_{k-1})^{2}b_{k-1}$ $\displaystyle c_{k}$
$\displaystyle=(1+c_{k-1})^{-2}c_{k-1}^{2}a_{k-1}$

hold; in this case we set $\varphi_{k}=R=\left(\begin{smallmatrix}1&0\\\
1&1\end{smallmatrix}\right)$.

By induction, it follows from the construction that
$\tau_{k}=\varphi_{1}\circ\varphi_{2}\circ\dots\circ\varphi_{k}(\tau_{0})$ for
every $k$. (Note that the order of the $\varphi_{i}$ is not necessarily the
one that might have been anticipated.)

In particular,
$\varphi(\tau_{0})=\tau_{k_{0}}=\varphi_{1}\circ\varphi_{2}\circ\dots\circ\varphi_{k_{0}}(\tau_{0})$.
Since the stabilizer of $\tau_{0}$ in the mapping class group
$\pi_{0}\,\mathrm{Diff}^{+}(S_{1,1})$ is generated by
$J=\left(\begin{smallmatrix}-1&0\\\ 0&-1\end{smallmatrix}\right)$, it follows
that

$\varphi=\varphi_{1}\circ\varphi_{2}\circ\dots\circ\varphi_{k_{0}}\circ
J^{\varepsilon}$

for some $\varepsilon\in\\{0,1\\}$. This concludes the proof. ∎

Proposition 13 associates to the edge weight system $(a_{k},b_{k},c_{k})$ and
to a puncture weight $h_{v}$ with $h_{v}^{n}=a_{k}^{2}b_{k}^{2}c_{k}^{2}$ an
irreducible representation
$\rho_{k}\colon\mathcal{T}_{\tau_{k}}^{q}(S)\to\mathrm{End}(V)$. Our
computation will require us to be more specific.

First of all, instead of the puncture invariant $h_{v}$, it is more convenient
to consider its square root $h$ uniquely determined by the properties that

$\displaystyle h^{2}$ $\displaystyle=h_{v}$ $\displaystyle
h^{n}=a_{0}b_{0}c_{0}.$

We begin with an arbitrary triple $(x_{0},y_{0},z_{0})$ with
$x_{0}^{n}=a_{0}$, $y_{0}^{n}=b_{0}$, $z_{0}^{n}=c_{0}$ and
$x_{0}y_{0}z_{0}=h$. Then, to take advantage of Lemma 25, we inductively
define $(x_{k},y_{k},z_{k})$ in the following way.

If $\varphi_{k}=L$, we set $u_{k}=qy_{k-1}$ and we pick an arbitrary
$v_{k}\in\mathbb{C}$ such that $v_{k}^{n}=1+u_{k}^{n}$. Then,

(3) $\displaystyle x_{k}$ $\displaystyle=y_{k-1}^{-1}$ $\displaystyle y_{k}$
$\displaystyle=v_{k}^{2}x_{k-1}$ $\displaystyle z_{k}$
$\displaystyle=v_{k}^{-2}y_{k-1}^{2}z_{k-1}.$

If $\varphi_{k}=R$, we set $u_{k}=qz_{k-1}$ and we again pick an arbitrary
$v_{k}\in\mathbb{C}$ such that $v_{k}^{n}=1+u_{k}^{n}$. Then,

(4) $\displaystyle x_{k}$ $\displaystyle=z_{k-1}^{-1}$ $\displaystyle y_{k}$
$\displaystyle=v_{k}^{2}y_{k-1}$ $\displaystyle z_{k}$
$\displaystyle=v_{k}^{-2}z_{k-1}^{2}x_{k-1}.$

Note that $(x_{k}^{n},y_{k}^{n},z_{k}^{n})=(a_{k},b_{k},c_{k})$, and that
$u_{k}=qx_{k}^{-1}$ and $v_{k}^{n}=1+a_{k}^{-1}$ in both cases. Also, the
product $x_{k}y_{k}z_{k}$ is independent of $k$, and is consequently equal to
the preferred square root $h$ of the puncture weight $h_{v}$.

Although $(a_{k_{0}},b_{k_{0}},c_{k_{0}})=(a_{0},b_{0},c_{0})$, there is no
reason for $(x_{k_{0}},y_{k_{0}},z_{k_{0}})$ to be equal to
$(x_{0},y_{0},z_{0})$. We just know that
$(x_{k_{0}}^{n},y_{k_{0}}^{n},z_{k_{0}}^{n})=(x_{0}^{n},y_{0}^{n},z_{0}^{n})$
and $x_{k_{0}}y_{k_{0}}z_{k_{0}}=x_{0}y_{0}z_{0}$. As a consequence, there
exists $l_{0}$, $m_{0}$, $n_{0}\in\mathbb{Z}$ with $l_{0}+m_{0}+n_{0}=0$ such
that

$\displaystyle x_{0}$ $\displaystyle=q^{4l_{0}}x_{k_{0}}$ $\displaystyle
y_{0}$ $\displaystyle=q^{4m_{0}}y_{k_{0}}$ $\displaystyle z_{0}$
$\displaystyle=q^{4n_{0}}z_{k_{0}}.$

######  Proposition 28.

Under the hypotheses of Theorem 16, the intertwiner $\Lambda_{\varphi,r}^{q}$
of Proposition 4 coincides, up to conjugation and rescaling by a factor with
modulus $1$, with the composition

$\Lambda=\Lambda_{1}\circ\Lambda_{2}\circ\dots\circ\Lambda_{k_{0}}\circ
T_{l_{0}m_{0}n_{0}}$

where

$\Lambda_{k}=\begin{cases}L_{u_{k}v_{k}}&\text{ if }\varphi_{k}=L\\\
R_{u_{k}v_{k}}&\text{ if }\varphi_{k}=R\end{cases}$

for the elementary intertwining operators $L_{uv}$, $R_{uv}$ and
$T_{l_{0}m_{0}n_{0}}\colon\mathbb{C}^{n}\to\mathbb{C}^{n}$ of §4.4 and §4.5,
and where the numbers $u_{k}$, $v_{k}\in\mathbb{C}^{*}$ and $k_{0}$, $l_{0}$,
$m_{0}\in\mathbb{Z}$ are defined as above.

###### Proof.

For each $k$, the edge weight system $(x_{k},y_{k},z_{k})$ defines a standard
representation
$\rho_{x_{k}y_{k}z_{k}}\colon\mathcal{T}^{q}\to\mathrm{End}(\mathbb{C}^{n})$
as in §4.2, and therefore a representation
$\rho_{k}=\rho_{x_{k}y_{k}z_{k}}\circ\theta_{\tau_{k}}^{-1}\colon\mathcal{T}_{\tau_{k}}^{q}(S)\to\mathrm{End}(\mathbb{C}^{n})$
by composition with the canonical isomorphism between $\mathcal{T}^{q}$ and
the Chekhov-Fock algebra of the ideal triangulation $\tau_{k}$.

Recall that $\tau_{k}$ is obtained from $\tau_{k-1}$ by a diagonal exchange
and an edge relabelling. The formulas of [Liu09] for the Chekhov-Fock
coordinate change isomorphisms in the special case of the one-puncture torus
then show that

$\theta_{\tau_{k-1}}^{-1}\circ\Phi_{\tau_{k-1}\tau_{k}}^{q}\circ\theta_{\tau_{k}}\colon\mathcal{T}^{q}\to\mathcal{T}^{q}$

coincides with the isomorphism $\mathcal{L}$ of §4.4 if the diagonal exchange
is performed along the edge $g_{k-1}$, and with $\mathcal{R}$ if the diagonal
exchange is performed along the edge $f_{k-1}$. Equivalently, this isomorphism
$\theta_{\tau_{k-1}}^{-1}\circ\Phi_{\tau_{k-1}\tau_{k}}^{q}\circ\theta_{\tau_{k}}$
is equal to $\mathcal{L}$ if $\varphi_{k}=L$, and to $\mathcal{R}$ if
$\varphi_{k}=R$.

Lemma 25 then shows that

$\rho_{k-1}\circ\Phi_{\tau_{k-1}\tau_{k}}^{q}(W)=\Lambda_{k}\circ\rho_{k}(W)\circ\Lambda_{k}^{-1}$

for every $W\in\mathcal{T}_{\tau_{k}}^{q}(S_{1,1})$.

Similarly, let
$\Psi_{\varphi,\tau_{0}}^{q}\colon\mathcal{T}_{\tau_{0}}^{q}(S_{1,1})\to\mathcal{T}_{\tau_{k_{0}}}^{q}(S_{1,1})$
be the isomorphism induced by $\varphi$, sending the generator
$\mathcal{T}_{\tau_{0}}^{q}(S_{1,1})$ corresponding to an edge of $\tau_{0}$
to the generator of $\mathcal{T}_{\tau_{k_{0}}}^{q}(S_{1,1})$ corresponding to
the image of that edge under $\varphi$. With our conventions,
$\Psi_{\varphi,\tau_{0}}^{q}$ is just
$\theta_{\tau_{k_{0}}}\circ\theta_{\tau_{0}}^{-1}$. Lemma 26 then shows that

$\rho_{k_{0}}\circ\Psi_{\varphi,\tau_{0}}^{q}(W)=T_{l_{0}m_{0}n_{0}}\circ\rho_{0}(W)\circ
T_{l_{0}m_{0}n_{0}}^{-1}.$

Combining these properties, we see that

$\rho_{0}\circ\Phi_{\tau_{k_{0}}\tau_{0}}^{q}\circ\Psi_{\varphi,\tau_{0}}^{q}(W)=\Lambda\circ\rho_{0}(W)\circ\Lambda^{-1}$

for the isomorphism $\Lambda\colon\mathbb{C}^{n}\to\mathbb{C}^{n}$ defined in
the statement.

This is the property satisfied by the intertwiner
$\bar{\Lambda}_{\varphi,\bar{r}}^{q}$ of Proposition 15. The determinant of
$\Lambda$ has modulus 1 by Lemmas 24 and 26. Proposition 15 then shows that
$\bar{\Lambda}_{\varphi,\bar{r}}^{q}$ and $\Lambda$ coincide up to conjugation
and rescaling by a modulus 1 scalar.

Theorem 16 then shows that the three intertwiners $\Lambda$,
$\bar{\Lambda}_{\varphi,\bar{r}}^{q}$ and $\Lambda_{\varphi,r}^{q}$ coincide
up to conjugation and modulus 1 rescaling. ∎

Note that there was some freedom in this computation, as we could arbitrarily
choose $n$–roots $v_{k}$ such that $v_{k}^{n}=1+u_{k}^{n}$. These choices are
balanced by the correction factors $l_{0}$, $m_{0}$, $n_{0}\in\mathbb{Z}$, in
the sense that different choices for the $v_{k}$ will in general lead to
different values for these correction factors.

We are of course interested in applying Proposition 28 in the context of
Conjecture 6. We now connect the two points of view.

In the hypotheses of that conjecture, we were given a character
$[r]\in\mathcal{X}_{\mathrm{SL}_{2}(\mathbb{C})}(S_{1,1})$ and a puncture
weight $\theta_{v}\in\mathbb{C}$ such that
$\operatorname{\mathrm{Trace}}r(\alpha_{v})=-\mathrm{e}^{\theta_{v}}-\mathrm{e}^{-\theta_{v}}$
when $\alpha_{v}$ is a small loop going once around the puncture. In the above
construction we were given, in addition to the periodic edge weight system
$(a_{0},b_{0},c_{0})$, $(a_{1},b_{1},c_{1})$, …,
$(a_{k_{0}},b_{k_{0}},c_{k_{0}})=(a_{0},b_{0},c_{0})$ defining the character
$[\bar{r}]\in\mathcal{X}_{\mathrm{PSL}_{2}(\mathbb{C})}(S_{1,1})$, a global
weight $h\in\mathbb{C}^{*}$ that is an $n$–root of the products
$a_{k}b_{k}c_{k}$ (which are independent of $k$).

######  Lemma 29.

With the above data, the hypotheses of Theorem 16 hold (and we can apply
Proposition 28) if and only if
$[r]\in\mathcal{X}_{\mathrm{SL}_{2}(\mathbb{C})}(S_{1,1})$ lifts
$[\bar{r}]\in\mathcal{X}_{\mathrm{PSL}_{2}(\mathbb{C})}(S_{1,1})$ and
$h=\mathrm{e}^{\pm\frac{1}{n}\theta_{v}}$.

###### Proof.

The puncture weight $h_{v}$ of Proposition 13 is related to $h$ by the
property that $h_{v}=h^{2}$. Then the relation $p_{v}^{2}=h_{v}+h_{v}^{-1}+2$
of Theorem 16 holds if and only if $p_{v}=\pm(h+h^{-1})$.

The sign $\pm$ is actually restricted by the geometry. Indeed, a simple
computation (see for instance [Bon09, §8.4]) shows that
$\operatorname{\mathrm{Trace}}r(\alpha_{v})=-a_{0}b_{0}c_{0}-(a_{0}b_{0}c_{0})^{-1}$.
Since
$\operatorname{\mathrm{Trace}}r(\alpha_{v})=-\mathrm{e}^{\theta_{v}}-\mathrm{e}^{-\theta_{v}}$
in the setup of Conjecture 6, it follows that
$\mathrm{e}^{\theta_{v}}=(a_{0}b_{0}c_{0})^{\pm 1}$. Because of the hypothesis
$p_{v}=\mathrm{e}^{\frac{1}{n}\theta_{v}}+\mathrm{e}^{-\frac{1}{n}\theta_{v}}$
in Conjecture 6 and because $h^{n}=a_{0}b_{0}c_{0}$, the condition
$p_{v}=\pm(h+h^{-1})$ is therefore equivalent to
$h=\mathrm{e}^{\pm\frac{1}{n}\theta_{v}}$ (and $p_{v}=+(h+h^{-1})$). ∎

###  4.7. Computing the correction factors $(l_{0},m_{0},n_{0})$

We need a practical way to compute the correction terms
$(l_{0},m_{0},n_{0})\in\mathbb{Z}^{3}$ that occurred in the previous section,
and this in a way that is systematic in $n$ so that we can better estimate
asymptotics in the framework of Conjecture 6. The general idea is that,
whenever we need to choose an $n$–root for a quantity $w\in\mathbb{Z}^{*}$, we
will take it of the form $\exp\frac{1}{n}W$ for a fixed “logarithm”
$W\in\mathbb{C}$, namely a number independent of $n$ such that $\exp W=w$. We
will try to systematically denote these logarithms by the capital letter
corresponding to the lower-case original data.

Conjecture 6 is unchanged if we replace $\theta_{v}$ by $-\theta_{v}$ in its
hypothesis. By Lemma 29, we can therefore arrange that
$h=\mathrm{e}^{\frac{1}{n}\theta_{v}}$.

Since $\mathrm{e}^{\theta_{v}}=h^{n}=a_{0}b_{0}c_{0}$, we can now choose
“logarithms” $A_{0}$, $B_{0}$, $C_{0}\in\mathbb{C}$ for $a_{0}$, $b_{0}$,
$c_{0}$, such that

$\displaystyle a_{0}$ $\displaystyle=\exp A_{0}$ $\displaystyle b_{0}$
$\displaystyle=\exp B_{0}$ $\displaystyle c_{0}$ $\displaystyle=\exp C_{0}$

and $A_{0}+B_{0}+C_{0}=\theta_{v}$.

After these initial choices, we inductively define $A_{k}$, $B_{k}$,
$C_{k}\in\mathbb{C}$ as follows. At each step of the induction, we arbitrarily
choose a “logarithm” $V_{k}\in\mathbb{C}$ such that

$\exp V_{k}=1+a_{k}^{-1}.$

We could systematically take $V_{k}=\log\left(1+a_{k}^{-1}\right)$ for
whatever (discontinuous) complex logarithm function $\log$ we prefer, but in
[BWY22b] it will be convenient to make these $V_{k}$ depend continuously on
the original data. We consequently allow here some flexibility in the choice
of the $V_{k}$.

We then set

$\displaystyle A_{k}$ $\displaystyle=-B_{k-1}$ $\displaystyle B_{k}$
$\displaystyle=2V_{k}+A_{k-1}$ $\displaystyle C_{k}$
$\displaystyle=-2V_{k}+2B_{k-1}+C_{k-1}$

if $\varphi_{k}=L$, and

$\displaystyle A_{k}$ $\displaystyle=-C_{k-1}$ $\displaystyle B_{k}$
$\displaystyle=2V_{k}+B_{k-1}$ $\displaystyle C_{k}$
$\displaystyle=-2V_{k}+2C_{k-1}+A_{k-1}.$

if $\varphi_{k}=R$.

The construction is designed so that, if for each $n$ we define

(5) $\displaystyle u_{k}$ $\displaystyle=\textstyle
q\exp\left(-\frac{1}{n}A_{k}\right)$ $\displaystyle v_{k}$
$\displaystyle=\textstyle\exp\frac{1}{n}V_{k}$ (6) $\displaystyle x_{k}$
$\displaystyle=\textstyle\exp\frac{1}{n}A_{k}$ $\displaystyle y_{k}$
$\displaystyle=\textstyle\exp\frac{1}{n}B_{k}$ $\displaystyle z_{k}$
$\displaystyle=\textstyle\exp\frac{1}{n}C_{k}$

these $u_{k}$, $v_{k}$, $x_{k}$, $z_{k}$, $z_{k}$ satisfy the required
relations (3–4).

In particular, $x_{k}^{n}=a_{k}$, $y_{k}^{n}=b_{k}$, $z_{k}^{n}=c_{k}$ and
$x_{k}y_{k}z_{k}=h=\exp\frac{1}{n}\theta_{v}$ for every $k$.

Since $(a_{k_{0}},b_{k_{0}},c_{k_{0}})=(a_{0},b_{0},c_{0})$ it follows that,
for the corresponding logarithms, there exists $\widehat{l}_{0}$,
$\widehat{m}_{0}$, $\widehat{n}_{0}\in\mathbb{Z}$ such that

$\displaystyle A_{0}$
$\displaystyle=A_{k_{0}}+2\pi\mathrm{i}\,\widehat{l}_{0}$ $\displaystyle
B_{0}$ $\displaystyle=B_{k_{0}}+2\pi\mathrm{i}\,\widehat{m}_{0}$
$\displaystyle C_{0}$
$\displaystyle=C_{k_{0}}+2\pi\mathrm{i}\,\widehat{n}_{0}.$

By construction, the quantity $A_{k}+B_{k}+C_{k}$ is independent of $k$. It
follows that

$\widehat{l}_{0}+\widehat{m}_{0}+\widehat{n}_{0}=0.$

######  Lemma 30.

If $q=\mathrm{e}^{\frac{2\pi\mathrm{i}}{n}}$ and for these choices of $u_{k}$,
$v_{k}$, $x_{k}$, $y_{k}$, $z_{k}$, we can take in the formula of Proposition
28 the correction factors $l_{0}$, $m_{0}$, $n_{0}\in\mathbb{Z}$ to be equal
to

$\displaystyle l_{0}$
$\displaystyle=\widehat{l}_{0}\textstyle\frac{(n-1)^{2}}{4}$ $\displaystyle
m_{0}$ $\displaystyle=\widehat{m}_{0}\textstyle\frac{(n-1)^{2}}{4}$
$\displaystyle n_{0}$
$\displaystyle=\widehat{n}_{0}\textstyle\frac{(n-1)^{2}}{4}$

where these quantities are defined as above.

###### Proof.

By construction,

$\displaystyle x_{0}$
$\displaystyle=\textstyle\exp\frac{1}{n}A_{0}=\textstyle\exp\left(\frac{1}{n}A_{k_{0}}+\frac{2\pi\mathrm{i}}{n}\,\widehat{l}_{0}\right)=q^{\widehat{l}_{0}}x_{k_{0}}=q^{4l_{0}}x_{k_{0}}$

since $q^{(n-1)^{2}}=q$. Similarly, $y_{0}=q^{4m_{0}}y_{k_{0}}$ and
$z_{0}=q^{4n_{0}}z_{k_{0}}$.

This is exactly what we needed to apply Proposition 28. ∎

######  Remark 31.

If we take the different value $q=\mathrm{e}^{\frac{2k\pi\mathrm{i}}{n}}$ for
some $k\in\mathbb{Z}$ coprime with $n$, then in the formulas of Lemma 30 the
terms $\frac{(n-1)^{2}}{4}$ will get replaced by the inverse of $4k$ modulo
$n$.

###  4.8. Explicit formulas

By Proposition 28, the intertwiner of Proposition 4 can be taken as

$\Lambda_{\varphi,r}^{q}=\Lambda_{1}\Lambda_{2}\dots\Lambda_{k_{0}}T_{l_{0}m_{0}n_{0}}$

for the isomorphisms $\Lambda_{k}$ and $T_{l_{0}m_{0}n_{0}}$ defined there. As
a consequence, its trace is equal to

$\operatorname{\mathrm{Trace}}\Lambda_{\varphi,r}^{q}=\sum_{i_{0},\,i_{1},\dots,\,i_{k_{0}}=1}^{n}(\Lambda_{1})_{i_{0}i_{1}}(\Lambda_{2})_{i_{1}i_{2}}\dots(\Lambda_{k_{0}})_{i_{k_{0}-1}i_{k_{0}}}(T_{l_{0}m_{0}n_{0}})_{i_{k_{0}}i_{0}}$

where $(\Lambda_{k})_{ij}$ and $(T_{l_{0}m_{0}n_{0}})_{ij}$ are the entries of
the matrices of the elementary intertwiners $\Lambda_{k}$,
$T_{l_{0}m_{0}n_{0}}\colon\mathbb{C}^{n}\to\mathbb{C}^{n}$.

In particular, for the discrete quantum dilogarithm function of §4.3,

$(\Lambda_{k})_{ij}=(L_{u_{k}v_{k}})_{ij}=\frac{{\mathrm{QDL}}^{q}(u_{k},v_{k}\,|\,2j)}{\left|D^{q}(u_{k})\right|^{\frac{1}{n}}\sqrt{n}}\,q^{-i^{2}+j^{2}+4ij+i-j}$

if the matrix $\varphi_{k}$ is equal to $L$, and

$(\Lambda_{k})_{ij}=(R_{u_{k}v_{k}})_{ij}=\frac{{\mathrm{QDL}}^{q}(u_{k},v_{k}\,|\,2j)}{\left|D^{q}(u_{k})\right|^{\frac{1}{n}}\sqrt{n}}\,q^{i^{2}+3j^{2}-4ij+i-j}$

if $\varphi_{k}=R$. We can combine both cases by saying that

$(\Lambda_{k})_{ij}=\frac{{\mathrm{QDL}}^{q}(u_{k},v_{k}\,|\,2j)}{\left|D^{q}(u_{k})\right|^{\frac{1}{n}}\sqrt{n}}\,q^{\varepsilon_{k}(i^{2}+j^{2}-4ij)+2j^{2}+i-j}$

with

$\varepsilon_{k}=\begin{cases}-1&\text{if }\varphi_{k}=L\\\ +1&\text{if
}\varphi_{k}=R.\end{cases}$

Also,

$(T_{l_{0}m_{0}n_{0}})_{ij}=q^{2j(n_{0}-m_{0})}\delta_{i,j+l_{0}}$

Therefore,

$\displaystyle\operatorname{\mathrm{Trace}}\Lambda_{\varphi,r}^{q}$
$\displaystyle=A_{n}^{\prime}\sum_{i_{0},\,i_{1},\dots,\,i_{k_{0}}=1}^{n}\prod_{k=1}^{k_{0}}{\mathrm{QDL}}^{q}(u_{k},v_{k}\,|\,2i_{k})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad
q^{\sum_{k=1}^{k_{0}}\varepsilon_{k}(i_{k-1}^{2}+i_{k}^{2}-4i_{k-1}i_{k})+2\sum_{k=1}^{k_{0}}i_{k}^{2}+i_{0}-i_{k_{0}}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
q^{2i_{0}(n_{0}-m_{0})}\delta_{i_{k_{0}},i_{0}+l_{0}}$
$\displaystyle=A_{n}\sum_{i_{1},\,i_{2},\dots,\,i_{k_{0}}=1}^{n}\prod_{k=1}^{k_{0}}{\mathrm{QDL}}^{q}(u_{k},v_{k}\,|\,2i_{k})$
$\displaystyle\qquad\qquad\qquad\qquad
q^{\varepsilon_{1}(i_{k_{0}}^{2}+i_{1}^{2}-4i_{k_{0}}i_{1})+\sum_{k=2}^{k_{0}}\varepsilon_{k}(i_{k-1}^{2}+i_{k}^{2}-4i_{k-1}i_{k})+2\sum_{k=1}^{k_{0}}i_{k}^{2}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
q^{4\varepsilon_{1}n_{0}i_{1}+2i_{k_{0}}(-\varepsilon_{1}kl_{0}-m_{0}+n_{0})}$

with

$A_{n}=\frac{q^{l_{0}(\varepsilon_{1}l_{0}+2m_{0}-2n_{0}-1)}}{n^{\frac{k_{0}}{2}}\prod_{k=1}^{k_{0}}\left|D^{q}(u_{k})\right|^{\frac{1}{n}}}$

(and $A_{n}^{\prime}=A_{n}q^{-l_{0}(\varepsilon_{1}l_{0}+2m_{0}-2n_{0}-1)}$).
If we set $\varepsilon_{k_{0}+1}=\varepsilon_{1}$, this can also be rewritten
as

$\displaystyle\operatorname{\mathrm{Trace}}\Lambda_{\varphi,r}^{q}$
$\displaystyle=A_{n}\sum_{i_{1},\,i_{2},\dots,\,i_{k_{0}}=1}^{n}\prod_{k=1}^{k_{0}}{\mathrm{QDL}}^{q}(u_{k},v_{k}\,|\,2i_{k})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad
q^{\sum_{k=1}^{k_{0}}i_{k}^{2}(\varepsilon_{k}+\varepsilon_{k+1}+2)-4\sum_{k=1}^{k_{0}-1}\varepsilon_{k+1}i_{k}i_{k+1}-4\varepsilon_{1}i_{k_{0}}i_{1}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
q^{4\varepsilon_{1}l_{0}i_{1}+2i_{k_{0}}(-\varepsilon_{1}l_{0}-m_{0}+n_{0})}.$

If we count indices modulo $k_{0}$, so that $i_{k_{0}+1}=i_{1}$, the third
line can be written in a more compact way by using

$-4\sum_{k=1}^{k_{0}-1}\varepsilon_{k+1}i_{k}i_{k+1}-4\varepsilon_{1}i_{k_{0}}i_{1}=-4\sum_{k=1}^{k_{0}}\varepsilon_{k+1}i_{k}i_{k+1}$

Finally, remember from Lemma 30 that we can take
$l_{0}=\widehat{l}_{0}\frac{(n-1)^{2}}{4}$,
$m_{0}=\widehat{m}_{0}\frac{(n-1)^{2}}{4}$ and
$n_{0}=\widehat{n}_{0}\frac{(n-1)^{2}}{4}$ for integers $\widehat{l}_{0}$,
$\widehat{m}_{0}$, $\widehat{n}_{0}\in\mathbb{Z}$ that are independent of $n$,
and such that $\widehat{l}_{0}+\widehat{m}_{0}+\widehat{n}_{0}=0$. Then the
last term of the above expression is equal to

$\displaystyle
q^{4\varepsilon_{1}l_{0}i_{1}+2i_{k_{0}}(-\varepsilon_{1}l_{0}-m_{0}+n_{0})}$
$\displaystyle=q^{\varepsilon_{1}\widehat{l}_{0}i_{1}+\frac{-\varepsilon_{1}\widehat{l}_{0}-\widehat{m}_{0}+\widehat{n}_{0}}{2}i_{k_{0}}}.$

Note that $-\varepsilon_{1}\widehat{l}_{0}-\widehat{m}_{0}+\widehat{n}_{0}$ is
even since $\widehat{l}_{0}+\widehat{m}_{0}+\widehat{n}_{0}=0$.

We summarize this computation in the following statement.

######  Proposition 32.

Let $\varphi\colon S_{1,1}\to S_{1,1}$ be an orientation preserving
diffeomorphism of the one-puncture torus. Let
$[r]\in\mathcal{X}_{\mathrm{SL}_{2}(\mathbb{C})}(S_{1,1})$ be a
$\varphi$–invariant character associated to a sequence of edge weight systems
$(a_{0},b_{0},c_{0})$, $(a_{1},b_{1},c_{1})$, …,
$(a_{k_{0}-1},b_{k_{0}-1},c_{k_{0}-1})$,$(a_{k_{0}},b_{k_{0}},c_{k_{0}})=(a_{0},b_{0},c_{0})$
as in Lemma 27. Finally, let $\theta_{v}\in\mathbb{C}$ be such that
$\mathrm{e}^{\theta_{v}}=a_{0}b_{0}c_{0}$ and, for an odd integer $n$, let
$\Lambda_{\varphi,r}^{q}$ be the intertwiner associated by Proposition 4 to
the data of $\varphi$, $[r]$, $q=\mathrm{e}^{\frac{2\pi\mathrm{i}}{n}}$ and
$p_{v}=\mathrm{e}^{\frac{1}{n}\theta_{v}}+\mathrm{e}^{-\frac{1}{n}\theta_{v}}$.

Then, up to multiplication by a scalar with modulus $1$, the trace of
$\Lambda_{\varphi,r}^{q}$ is equal to

$\displaystyle\operatorname{\mathrm{Trace}}\Lambda_{\varphi,r}^{q}$
$\displaystyle=\frac{1}{n^{\frac{k_{0}}{2}}\prod_{k=1}^{k_{0}}\left|D^{q}(u_{k})\right|^{\frac{1}{n}}}$
$\displaystyle\qquad\qquad\sum_{i_{1},\,i_{2},\dots,\,i_{k_{0}}=1}^{n}\prod_{k=1}^{k_{0}}{\mathrm{QDL}}^{q}(u_{k},v_{k}\,|\,2i_{k})$
$\displaystyle\qquad\qquad\qquad\qquad
q^{\sum_{k=1}^{k_{0}}i_{k}^{2}(\varepsilon_{k}+\varepsilon_{k+1}+2)-4\sum_{k=1}^{k_{0}}\varepsilon_{k+1}i_{k}i_{k+1}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad
q^{\varepsilon_{1}\widehat{l}_{0}i_{1}+\frac{-\varepsilon_{1}\widehat{l}_{0}-\widehat{m}_{0}+\widehat{n}_{0}}{2}i_{k_{0}}}$

where the quantities $u_{k}$, $v_{k}$, $\varepsilon_{k}$, $\widehat{l}_{0}$,
$\widehat{m}_{0}$, $\widehat{n}_{0}$ are defined in §4.6 and §4.7 and where,
for $u$, $v\in\mathbb{C}$ with $v^{n}=1+u^{n}$,

${\mathrm{QDL}}^{q}(u,v\,|\,i)=v^{-i}\prod_{j=1}^{i}(1+uq^{-2j})$

and

$D^{q}(u)=\prod_{i=1}^{n}{\mathrm{QDL}}^{q}(u,v\,|\,i)=(1+u^{n})^{-\frac{n+1}{2}}\prod_{j=1}^{n}(1+uq^{-2j})^{n-j+1}.\qed$

###  4.9. A few examples

The algorithm developed in §§4.6–4.8 has many moving pieces. It is probably
useful to illustrate its implementation, and its subtleties, by applying it to
a few examples. The following ones all correspond to the diffeomorphism
$\varphi=LLR=\left(\begin{smallmatrix}3&2\\\ 1&1\end{smallmatrix}\right)$, and
actually correspond to the three cases that we encountered in the numerical
experiments of §2.

Since the diffeomorphism $\varphi$ is equal to $LLR$ in all three cases, we
have that $k_{0}=3$, $\varepsilon_{1}=-1$, $\varepsilon_{2}=-1$ and
$\varepsilon_{3}=+1$. Therefore,

(7) $\displaystyle\operatorname{\mathrm{Trace}}\Lambda_{\varphi,r}^{q}$
$\displaystyle=\frac{1}{n^{\frac{3}{2}}\left|D^{q}(u_{1})\right|^{\frac{1}{n}}\left|D^{q}(u_{2})\right|^{\frac{1}{n}}\left|D^{q}(u_{3})\right|^{\frac{1}{n}}}$
$\displaystyle\qquad\sum_{i_{1},\,i_{2},\,i_{3}=1}^{n}{\mathrm{QDL}}^{q}(u_{1},v_{1}\,|\,2i_{1}){\mathrm{QDL}}^{q}(u_{2},v_{2}\,|\,2i_{2}){\mathrm{QDL}}^{q}(u_{3},v_{3}\,|\,2i_{3})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad
q^{2i_{2}^{2}+2i_{3}^{2}+4i_{1}i_{2}-4i_{2}i_{3}+4i_{3}i_{1}}q^{-\widehat{l}_{0}i_{1}+\frac{\widehat{l}_{0}-\widehat{m}_{0}+\widehat{n}_{0}}{2}i_{3}}$

where the numbers $u_{k}$, $v_{k}\in\mathbb{C}$ and the integers
$\widehat{l}_{0}$, $\widehat{m}_{0}$, $\widehat{n}_{0}\in\mathbb{Z}$ are
determined by the algebraic data and by choices of logarithms. We now explain
how these parameters are chosen, according to the case considered.

First of all, since $\varphi=LLR$, the edge weights associated to the
corresponding ideal triangulation sweep are such that

$\displaystyle a_{1}$ $\displaystyle=b_{0}^{-1}$ $\displaystyle b_{1}$
$\displaystyle=(1+b_{0})^{2}a_{0}$ $\displaystyle c_{1}$
$\displaystyle=(1+b_{0})^{-2}b_{0}^{2}c_{0}$ $\displaystyle a_{2}$
$\displaystyle=b_{1}^{-1}$ $\displaystyle b_{2}$
$\displaystyle=(1+b_{1})^{2}a_{1}$ $\displaystyle c_{2}$
$\displaystyle=(1+b_{1})^{-2}b_{1}^{2}c_{1}$ $\displaystyle a_{3}$
$\displaystyle=c_{2}^{-1}$ $\displaystyle b_{3}$
$\displaystyle=(1+c_{2})^{2}b_{2}$ $\displaystyle c_{3}$
$\displaystyle=(1+c_{2})^{-2}c_{2}^{2}a_{2}$ $\displaystyle a_{0}$
$\displaystyle=a_{3}$ $\displaystyle b_{0}$ $\displaystyle=b_{3}$
$\displaystyle c_{0}$ $\displaystyle=c_{3}.$

In general, we can only rely on numerical solutions to this system of
equations, but it turns out that this example is still sufficiently small that
we can explicitly find all solutions by an elementary algebraic manipulation.
Namely, the initial edge weights $(a_{0},b_{0},c_{0})$ of these solutions are
of the form

$\displaystyle b_{0}$
$\displaystyle=-\frac{2a_{0}+1}{2(a_{0}+1)}\pm\mathrm{i}\frac{\sqrt{8a_{0}^{2}+11a_{0}+4}}{2(a_{0}+1)\sqrt{a_{0}}}\text{
or
}-\frac{2a_{0}+3}{2(a_{0}+1)}\pm\mathrm{i}\frac{\sqrt{3a_{0}+4}}{2(a_{0}+1)\sqrt{a_{0}}}$
$\displaystyle c_{0}$
$\displaystyle=\frac{(1+a_{0}(1+b_{0})^{2})^{2}}{a_{0}^{3}b_{0}^{2}(1+b_{0})^{2}},$

excluding in each case finitely many values for $a_{0}$ to guarantee that all
edge weights $a_{k}$, $b_{k}$, $c_{k}$ are different from 0. In particular,
the space of solutions consists of two connected complex curves in
$\big{(}\mathbb{C}^{*}\big{)}^{3}$, which are disjoint by another computation.

The hyperbolic character
$[\bar{r}_{\mathrm{hyp}}]\in\mathcal{X}_{\mathrm{PSL}_{2}(\mathbb{C})}(S_{1,1})$
corresponding to the monodromy of the complete hyperbolic metric of the
mapping torus $M_{\varphi}$ is associated to the initial edge weight system
$(a_{0},b_{0},c_{0})=\left(\frac{-1-\mathrm{i}\sqrt{7}}{4},\frac{-3+\mathrm{i}\sqrt{7}}{2},\frac{5-\mathrm{i}\sqrt{7}}{8}\right)$,
which is in the curve defined by the first solution for $b_{0}$

Example 1. We consider the case where

$\displaystyle a_{0}$ $\displaystyle=-0.75-0.1\,\mathrm{i}$ $\displaystyle
b_{0}$
$\displaystyle=-\frac{2a_{0}+1}{2(a_{0}+1)}-\mathrm{i}\frac{\sqrt{8a_{0}^{2}+11a_{0}+4}}{2(a_{0}+1)\sqrt{a_{0}}}\approx
1.51712+1.20930\,\mathrm{i}$ $\displaystyle c_{0}$
$\displaystyle=\frac{(1+a_{0}(1+b_{0})^{2})^{2}}{a_{0}^{3}b_{0}^{2}(1+b_{0})^{2}}\approx-2.25462+0.617377\,\mathrm{i}$

(where $\sqrt{z}$ is the usual complex square root, with
$-\frac{\pi}{2}<\operatorname{\mathfrak{Im}}\sqrt{\leqslant}\frac{\pi}{2}$).
This is the example considered for the numerical data of Figure 2, largely
selected for aesthetic reasons. It defines a $\varphi$–invariant hyperbolic
character $[\bar{r}]\in\mathcal{X}_{\mathrm{PSL}_{2}(\mathbb{C})}(S_{1,1})$
which is in the same component of the fixed point set of the action of
$\varphi$ as the hyperbolic character $[\bar{r}_{\mathrm{hyp}}]$. In
particular, it lifts to a character
$[r]\in\mathcal{X}_{\mathrm{SL}_{2}(\mathbb{C})}(S_{1,1})$, and in facts
admits exactly two such lifts since the fixed point set of the action of
$\varphi$ on $H^{1}(S_{1,1};\mathbb{Z}/2)$ is isomorphic to $\mathbb{Z}/2$.

The first diagram of Figure 2 corresponds to the case where, for the standard
complex logarithm function $\log$, the puncture invariant $\theta_{v}$ is
equal to

$\theta_{v}=\log a_{0}+\log b_{0}+\log c_{0}.$

Then, if we follow the algorithm of §4.7, we can choose

$\displaystyle A_{0}$ $\displaystyle=\log a_{0}$ $\displaystyle B_{0}$
$\displaystyle=\log b_{0}$ $\displaystyle C_{0}$ $\displaystyle=\log c_{0}$
$\displaystyle\approx-0.278871-3.00904\,\mathrm{i}$ $\displaystyle\approx
0.662750+0.672970\,\mathrm{i}$ $\displaystyle\approx
0.849133+2.87432\,\mathrm{i}$ $\displaystyle A_{1}$ $\displaystyle=-B_{0}$
$\displaystyle B_{1}$ $\displaystyle=2\log(1+a_{1}^{-1})+A_{0}$ $\displaystyle
C_{1}$ $\displaystyle=-2\log(1+a_{1}^{-1})+2B_{0}+C_{0}$ $\displaystyle A_{2}$
$\displaystyle=-B_{1}$ $\displaystyle B_{2}$
$\displaystyle=2\log(1+a_{2}^{-1})+A_{1}$ $\displaystyle C_{2}$
$\displaystyle=-2\log(1+a_{2}^{-1})+2B_{1}+C_{1}$ $\displaystyle A_{3}$
$\displaystyle=-C_{2}$ $\displaystyle B_{3}$
$\displaystyle=2\log(1+a_{3}^{-1})+B_{2}$ $\displaystyle C_{3}$
$\displaystyle=-2\log(1+a_{3}^{-1})+2C_{2}+A_{2}$
$\displaystyle\approx-0.278871-3.00904\,\mathrm{i}$ $\displaystyle\approx
0.662750+0.672970\,\mathrm{i}$ $\displaystyle\approx
0.849133+2.87432\,\mathrm{i}$

by systematically selecting $V_{k}=\log(1+a_{k}^{-1})$.

Since we abstractly know that

$\displaystyle A_{0}$ $\displaystyle=A_{3}+2\pi\mathrm{i}\,\widehat{l}_{0}$
$\displaystyle B_{0}$ $\displaystyle=B_{3}+2\pi\mathrm{i}\,\widehat{m}_{0}$
$\displaystyle C_{0}$ $\displaystyle=C_{3}+2\pi\mathrm{i}\,\widehat{n}_{0}.$

for some integers $\widehat{l}_{0}$, $\widehat{m}_{0}$,
$\widehat{n}_{0}\in\mathbb{Z}$, we numerically see that
$\widehat{l}_{0}=\widehat{m}_{0}=\widehat{n}_{0}=0$.

Therefore, in this example, the trace of the intertwiner
$\Lambda_{\varphi,r}^{q}$ is obtained by setting

$\displaystyle u_{1}$ $\displaystyle=\exp\frac{2\pi\mathrm{i}-A_{1}}{n}$
$\displaystyle u_{2}$ $\displaystyle=\exp\frac{2\pi\mathrm{i}-A_{2}}{n}$
$\displaystyle u_{3}$ $\displaystyle=\exp\frac{2\pi\mathrm{i}-A_{3}}{n}$
$\displaystyle v_{1}$ $\displaystyle=\exp\frac{\log(1+a_{1}^{-1})}{n}$
$\displaystyle v_{2}$ $\displaystyle=\exp\frac{\log(1+a_{2}^{-1})}{n}$
$\displaystyle v_{3}$ $\displaystyle=\exp\frac{\log(1+a_{3}^{-1})}{n}$
$\displaystyle\widehat{l}_{0}$ $\displaystyle=0$
$\displaystyle\widehat{m}_{0}$ $\displaystyle=0$
$\displaystyle\widehat{n}_{0}$ $\displaystyle=0$

in Equation (7).

Example 2. In the previous example, we can replace the puncture weight
$\theta_{v}$ by

$\theta^{\prime}_{v}=\theta_{v}+2\pi\mathrm{i}\eta$

for some integer $\eta\in\mathbb{Z}$, while keeping the same periodic edge
weight system $(a_{0},b_{0},c_{0})$, $(a_{1},b_{1},c_{1})$,
$(a_{2},b_{2},c_{2})$, $(a_{3},b_{3},c_{3})=(a_{0},b_{0},c_{0})$ for the ideal
triangulation sweep. For instance, the second diagram of Figure 2 corresponds
to the case where $\eta=1$.

Then, we can replace the logarithms $A_{0}$, $B_{0}$, $C_{0}$ of the previous
case with

$\displaystyle A_{0}^{\prime}$ $\displaystyle=A_{0}+2\pi\mathrm{i}\eta$
$\displaystyle B_{0}^{\prime}$ $\displaystyle=B_{0}$ $\displaystyle
C_{0}^{\prime}$ $\displaystyle=C_{0}$

in order to satisfy the relation
$A_{0}^{\prime}+B_{0}^{\prime}+C_{0}^{\prime}=\theta_{v}^{\prime}$. The
algorithm of §4.7 then gives

$\displaystyle A_{1}^{\prime}$ $\displaystyle=-B_{0}^{\prime}$ $\displaystyle
B_{1}^{\prime}$ $\displaystyle=2\log(1+a_{1}^{-1})+A_{0}^{\prime}$
$\displaystyle C_{1}^{\prime}$
$\displaystyle=-2\log(1+a_{1}^{-1})+2B_{0}^{\prime}+C_{0}^{\prime}$
$\displaystyle=A_{1}$ $\displaystyle=B_{1}+2\pi\mathrm{i}\eta$
$\displaystyle=C_{1}$ $\displaystyle A_{2}^{\prime}$
$\displaystyle=-B_{1}^{\prime}$ $\displaystyle B_{2}^{\prime}$
$\displaystyle=2\log(1+a_{2}^{-1})+A_{1}^{\prime}$ $\displaystyle
C_{2}^{\prime}$
$\displaystyle=-2\log(1+a_{2}^{-1})+2B_{1}^{\prime}+C_{1}^{\prime}$
$\displaystyle=A_{2}-2\pi\mathrm{i}\eta$ $\displaystyle=B_{2}$
$\displaystyle=C_{2}+4\pi\mathrm{i}\eta$ $\displaystyle A_{3}^{\prime}$
$\displaystyle=-C_{2}^{\prime}$ $\displaystyle B_{3}^{\prime}$
$\displaystyle=2\log(1+a_{3}^{-1})+B_{2}^{\prime}$ $\displaystyle
C_{3}^{\prime}$
$\displaystyle=-2\log(1+a_{3}^{-1})+2C_{2}^{\prime}+A_{2}^{\prime}$
$\displaystyle=A_{3}-4\pi\mathrm{i}\eta$ $\displaystyle=B_{3}$
$\displaystyle=C_{3}+6\pi\mathrm{i}\eta.$

It follows that in this more general example the parameters $u_{k}$, $v_{k}$,
$\widehat{l}_{0}$, $\widehat{m}_{0}$ and $\widehat{n}_{0}$ get replaced by

$\displaystyle u_{1}^{\prime}$
$\displaystyle=\exp\frac{2\pi\mathrm{i}-A_{1}^{\prime}}{n}=u_{1}$
$\displaystyle u_{2}^{\prime}$
$\displaystyle=\exp\frac{2\pi\mathrm{i}-A_{2}^{\prime}}{n}=q^{-\eta}u_{2}$
$\displaystyle u_{3}^{\prime}$
$\displaystyle=\exp\frac{2\pi\mathrm{i}-A_{3}^{\prime}}{n}=q^{2\eta}u_{3}$
$\displaystyle v_{1}^{\prime}$
$\displaystyle=\exp\frac{\log(1+a_{1}^{-1})}{n}=v_{1}$ $\displaystyle
v_{2}^{\prime}$ $\displaystyle=\exp\frac{\log(1+a_{2}^{-1})}{n}=v_{2}$
$\displaystyle v_{3}^{\prime}$
$\displaystyle=\exp\frac{\log(1+a_{3}^{-1})}{n}=v_{3}$
$\displaystyle\widehat{l}_{0}^{\prime}$ $\displaystyle=3\eta$
$\displaystyle\widehat{m}_{0}^{\prime}$ $\displaystyle=0$
$\displaystyle\widehat{n}_{0}^{\prime}$ $\displaystyle=-3\eta.$

Example 3. This time, we keep $\varphi=LLR$ and the corresponding ideal
triangulation sweep, but we choose a periodic edge weight system for the sweep
whose initial edge weight system
$(a_{0}^{\prime\prime},b_{0}^{\prime\prime},c_{0}^{\prime\prime})$ is given by

$\displaystyle a_{0}^{\prime\prime}$ $\displaystyle=1+\,\mathrm{i}$
$\displaystyle b_{0}^{\prime\prime}$
$\displaystyle=-\frac{2a_{0}^{\prime\prime}+3}{2(a_{0}^{\prime\prime}+1)}-\mathrm{i}\frac{\sqrt{3a_{0}^{\prime\prime}+4}}{2(a_{0}^{\prime\prime}+1)\sqrt{a_{0}^{\prime\prime}}}\approx-1.51564-0.311861\,\mathrm{i}$
$\displaystyle c_{0}^{\prime\prime}$
$\displaystyle=\frac{(1+a_{0}^{\prime\prime}(1+b_{0}^{\prime\prime})^{2})^{2}}{a_{0}^{\prime\prime
3}b_{0}^{\prime\prime
2}(1+b_{0}^{\prime\prime})^{2}}\approx-0.367012-0.130244\,\mathrm{i}$

A subtlety here is that it can be shown that the character
$[\bar{r}]\in\mathcal{X}_{\mathrm{PSL}_{2}(\mathbb{C})}(S_{1,1})$ defined by
this edge weight system does not lift to a $\varphi$–invariant
$\mathrm{SL}_{2}(\mathbb{C})$–character
$[r]\in\mathcal{X}_{\mathrm{SL}_{2}(\mathbb{C})}(S_{1,1})$. What is given by
the formula (7) is therefore, in this case, the trace of the intertwiner
$\bar{\Lambda}_{\varphi,\bar{r}}^{q}$ associated by Proposition 15 to the
puncture invariant $h_{v}=\mathrm{e}^{\frac{2}{n}\theta_{v}}$, where
$\theta_{v}$ is chosen so that
$\mathrm{e}^{\theta_{v}}=a_{0}^{\prime\prime}b_{0}^{\prime\prime}c_{0}^{\prime\prime}$.

This is the example illustrated in Figure 3, for $\theta_{v}=\log
a_{0}^{\prime\prime}+\log b_{0}^{\prime\prime}+\log c_{0}^{\prime\prime}$.
Then, if we start with

$\displaystyle A_{0}^{\prime\prime}$ $\displaystyle=\log a_{0}^{\prime\prime}$
$\displaystyle B_{0}^{\prime\prime}$ $\displaystyle=\log b_{0}^{\prime\prime}$
$\displaystyle C_{0}^{\prime\prime}$ $\displaystyle=\log c_{0}^{\prime\prime}$
$\displaystyle\approx 0.346574+0.785398\,\mathrm{i}$ $\displaystyle\approx
0.436571-2.93866\,\mathrm{i}$
$\displaystyle\approx-0.943051-2.80058\,\mathrm{i},$

the same computations as above now lead us to

$\displaystyle A_{3}^{\prime\prime}$ $\displaystyle\approx
0.346574+13.3518\,\mathrm{i}$ $\displaystyle B_{3}^{\prime\prime}$
$\displaystyle\approx 0.436571+3.34452\,\mathrm{i}$ $\displaystyle
C_{3}^{\prime\prime}$ $\displaystyle\approx-0.943051-21.6501\,\mathrm{i},$

and the correction factors are therefore

$\displaystyle\widehat{l}_{0}^{\prime\prime}$ $\displaystyle=-2$
$\displaystyle\widehat{m}_{0}^{\prime\prime}$ $\displaystyle=-1$
$\displaystyle\widehat{n}_{0}^{\prime\prime}$ $\displaystyle=+3$

At this point, it may be hard to understand why the algebraic data of this
example is so different from that of Examples 1 and 2 that, as observed in §2,
it leads to fundamentally distinct asymptotic behavior for the trace of the
associated intertwiner $\Lambda_{\varphi,\bar{r}}^{q}$. The cancellations that
occur in this case, and not in the two other examples, will be explained in
[BWY22b], where a specific criterion is developed. In the particular case of
the diffeomorphism $\varphi=LLR$, this criterion is based on the parity of
$\frac{\widehat{l}_{0}-\widehat{m}_{0}+\widehat{n}_{0}}{2}$.

##  5\. The case of a general surface

In addition to the fact that they are used in [BWY22a, BWY22b], our
computations in the case of the one-puncture torus were meant as a
demonstration project to show the effectiveness of the methods of §3 to
compute the intertwiner $\Lambda_{\varphi,r}^{q}$ of Proposition 4. The same
methods can be implemented for a general surface, but at the expense of
greatly increased computational complexity. We briefly describe how.

Our data is an orientation-preserving diffeomorphism $\varphi\colon S\to S$ of
a surface $S$ with genus $g$ and $p$ punctures. We are also given a
$\varphi$–invariant character
$[r]\in\mathcal{X}_{\mathrm{SL}_{2}(\mathbb{C})}(S)$ associated, as in §3.2,
to an edge weight system for an ideal triangulation sweep, and
$\varphi$–invariant puncture weights as in Theorem 2. We want to compute the
intertwiner $\Lambda_{\varphi,r}^{q}$ associated by Proposition 4 to this
data, by relying on Theorem 16 and computing instead the intertwiner
$\bar{\Lambda}_{\varphi,r}^{q}$ associated to suitably compatible data by
Proposition 15.

Our strategy requires some preparation for each surface $S$, after which it is
easily implemented for each diffeomorphism $\varphi\colon S\to S$ and each
ideal triangulation sweep $\tau=\tau^{(0)}$, $\tau^{(1)}$, …,
$\tau^{(k_{0}-1)}$, $\tau^{(k_{0})}=\varphi(\tau)$. We divide this
implementation into several steps.

For this, we fix a surface $S$ with genus $g$ and $p$ punctures. In
particular, every ideal triangulation of $S$ has $e=6g+3p-6$ edges.

Step 1. The first observation is that, up to orientation-preserving
diffeomorphism, the surface $S$ admits only finitely many ideal triangulations
$\tau$, which consequently gives us finitely many Chekhov-Fock algebras
$\mathcal{T}_{\tau}^{q}(S)$. For each such ideal triangulation $\tau$, one
needs a practical implementation of Proposition 13, associating a
representation
$\rho\colon\mathcal{T}_{\tau}^{q}(S)\to\mathrm{End}(\mathbb{C}^{n(3g+p-3)})$
to the data of edge weights $a_{i}\in\mathbb{C}^{*}$ and compatible puncture
weights $h_{v}\in\mathbb{C}^{*}$ for $\tau$.

In practice, this requires the effective block diagonalization over the
integers of the antisymmetric bilinear form
$\mathbb{Z}^{e}\times\mathbb{Z}^{e}\to\mathbb{Z}$ defined by the matrix whose
entries are the coefficients $\sigma_{ij}\in\\{-2,-1,0,1,2\\}$ occurring in
the $q$–commutativity relations $X_{i}X_{j}=q^{2ij}X_{j}X_{i}$ defining the
Chekhov-Fock algebra $\mathcal{T}_{\tau}^{q}$. Proposition 5 of [BL07]
predicts that there will be $g$ blocks $\left(\begin{smallmatrix}0&-2\\\
2&0\end{smallmatrix}\right)$, $2g+p-3$ blocks
$\left(\begin{smallmatrix}0&-1\\\ 1&0\end{smallmatrix}\right)$ and $p$ blocks
$\left(\begin{smallmatrix}0\end{smallmatrix}\right)$. In practice, this block
diagonalization is easily obtained by the usual Gram-Schmidt methods.

Once the $q$–commutativity relations are block diagonalized, the practical
implementation of Proposition 13 involves a choice of $n$–roots
$a_{i}^{\frac{1}{n}}$ for the edge weights $a_{i}\in\mathbb{C}^{*}$
constrained by the puncture weights $h_{v}$ in the sense that
$h_{v}=a_{i_{1}}^{\frac{1}{n}}a_{i_{2}}^{\frac{1}{n}}\dots
a_{i_{k}}^{\frac{1}{n}}$ whenever the puncture $v$ is adjacent to the edges
$\gamma_{i_{1}}$, $\gamma_{i_{2}}$, …, $\gamma_{i_{k}}$. The construction
associates to the $n$–roots $a_{i}^{\frac{1}{n}}$ a “standard” representation
$\rho\colon\mathcal{T}_{\tau}^{q}(S)\to\mathrm{End}(\mathbb{C}^{n(3g+p-3)})$.

In our discussion of the case of the one-puncture torus, this step essentially
corresponds to Proposition 21. In the general case as in this special case,
the image $\rho(X_{i})\in\mathrm{End}(\mathbb{C}^{n(3g+p-3)})$ of each
generator $X_{i}$ of $\mathcal{T}_{\tau}^{q}(S)$ is obtained by multiplying a
fixed matrix (depending only on $q$ and on the combinatorics of $\tau$) by a
certain monomial in the $n$–roots $a_{i}^{\frac{1}{n}}$.

Step 2. The previous step associates a representation
$\rho\colon\mathcal{T}_{\tau}^{q}(S)\to\mathrm{End}(\mathbb{C}^{n(3g+p-3)})$
to each suitable choice of $n$–roots $a_{i}^{\frac{1}{n}}$. Another choice of
$n$–roots will be of the form $q^{m_{i}}a_{i}^{\frac{1}{n}}$ for integers
$m_{i}\in\mathbb{Z}$, and will define a representation
$\rho^{\prime}\colon\mathcal{T}_{\tau}^{q}(S)\to\mathrm{End}(\mathbb{C}^{n(3g+p-3)})$
that is isomorphic to $\rho$ by an isomorphism $L_{m_{1}m_{2}\dots
m_{e}}\colon\mathbb{C}^{n(3g+p-3)}\to\mathbb{C}^{n(3g+p-3)}$. Namely,

$\rho(W)=L_{m_{1}m_{2}\dots m_{e}}\circ\rho^{\prime}(W)\circ
L_{m_{1}m_{2}\dots m_{e}}^{-1}\in\mathrm{End}(\mathbb{C}^{n(3g+p-3)})$

for every $W\in\mathcal{T}_{\tau}^{q}(S)$.

Once we are given the block diagonalization of the previous step, the
isomorphism $L_{m_{1}m_{2}\dots m_{e}}$ can be easily computed (up to scalar
multiplication) and looks very much like the twist isomorphism
$T_{l_{0}m_{0}n_{0}}$ of §4.5. In particular, when $S$ is the one-puncture
torus, the isomorphism $L_{m_{1}m_{2}m_{3}}$ can be chosen equal to
$T_{m_{1}^{\prime}m_{2}^{\prime}m_{3}^{\prime}}$ with
$m_{i}=4m_{i}^{\prime}\negthickspace\mod n$.

Choose a normalization such that $\left|\det L_{m_{1}m_{2}\dots
m_{e}}\right|=1$.

Step 3. For a given ideal triangulation $\tau$, there are finitely many ideal
triangulations $\tau^{\prime}$ that are obtained from $\tau$ by a diagonal
exchange or an edge relabelling. We saw in §3.2 that, at least under a
genericity condition guaranteeing that we do not try to divide by 0 in the
case of a diagonal exchange, an edge weight system
$a\in(\mathbb{C}^{*}\big{)}^{e}$ for $\tau$ determines an edge weight system
$a^{\prime}\in(\mathbb{C}^{*}\big{)}^{e}$ for $\tau^{\prime}$ that is
associated to the same character in
$\mathcal{X}_{\mathrm{PSL}_{2}(\mathbb{C})}(S)$. The gist of Proposition 14,
namely Lemma 27 of [BL07], asserts the following: If
$\rho\colon\mathcal{T}_{\tau}^{q}(S)\to\mathrm{End}(\mathbb{C}^{n(3g+p-3)})$
and
$\rho^{\prime}\colon\mathcal{T}_{\tau^{\prime}}^{q}(S)\to\mathrm{End}(\mathbb{C}^{n(3g+p-3)})$
are two standard representations respectively associated to systems of
$n$–roots $a_{i}^{\frac{1}{n}}$ and $a_{i}^{\prime\frac{1}{n}}$ compatible
with the same puncture invariants $h_{v}=h_{v}^{\prime}$, then, for the
Chekhov-Fock coordinate change
$\Phi_{\tau\tau^{\prime}}^{q}\colon\mathcal{T}_{\tau^{\prime}}^{q}(S)\to\mathcal{T}_{\tau}^{q}(S)$,
the representation
$\rho\circ\Phi_{\tau\tau^{\prime}}^{q}\colon\mathcal{T}_{\tau^{\prime}}^{q}(S)\to\mathrm{End}(\mathbb{C}^{n(3g+p-3)})$
makes sense and is isomorphic to $\rho^{\prime}$ by an isomorphism
$\Lambda\colon\mathbb{C}^{n(3g+p-3)}\to\mathbb{C}^{n(3g+p-3)}$. More
precisely,

$\rho\circ\Phi_{\tau\tau^{\prime}}^{q}(W)=\Lambda\circ\rho^{\prime}(W^{\prime})\circ\Lambda^{-1}\in\mathrm{End}(\mathbb{C}^{n(3g+p-3)})$

for every $W^{\prime}\in\mathcal{T}_{\tau^{\prime}}^{q}(S)$. Note that, once
we know it exists, the isomorphism $\Lambda$ can be explicitly determined by
solving the finitely many linear equations
$\Lambda\circ\big{(}\rho\circ\Phi_{\tau\tau^{\prime}}^{q}(X_{i}^{\prime})\big{)}=\Lambda\circ\rho^{\prime}(X_{i}^{\prime})$
given by the generators $X_{i}^{\prime}$ of
$\mathcal{T}_{\tau^{\prime}}^{q}(S)$.

In this Step 3, we first select a rule that, given a choice of $n$–roots
$a_{i}^{\frac{1}{n}}$ for an edge weight system
$a\in\big{(}\mathbb{C}^{*}\big{)}^{e}$ for $\tau$, specifies $n$–roots
$a_{i}^{\prime\frac{1}{n}}$ for the associated edge weight system
$a^{\prime}\in\big{(}\mathbb{C}^{*}\big{)}^{e}$ for $\tau^{\prime}$, in such a
way that these $n$–roots define the same puncture weights
$h_{v}=h_{v}^{\prime}$ in the sense that

$a_{i_{1}}^{\frac{1}{n}}a_{i_{2}}^{\frac{1}{n}}\dots
a_{i_{k}}^{\frac{1}{n}}=a_{i_{1}^{\prime}}^{\prime\frac{1}{n}}a_{i_{2}^{\prime}}^{\prime\frac{1}{n}}\dots
a_{i_{k^{\prime}}^{\prime}}^{\prime\frac{1}{n}}\ (=h_{v}=h_{v}^{\prime})$

whenever the puncture $v$ is adjacent to the edges $\gamma_{i_{1}}$,
$\gamma_{i_{2}}$, …, $\gamma_{i_{k}}$ of $\tau$ and to the edges
$\gamma_{i_{1}^{\prime}}^{\prime}$, $\gamma_{i_{2}^{\prime}}^{\prime}$, …,
$\gamma_{i_{k^{\prime}}^{\prime}}^{\prime}$ of $\tau^{\prime}$. This rule may
not necessarily be continuous.

Then, once the rule is fixed,
$\rho\colon\mathcal{T}_{\tau}^{q}(S)\to\mathrm{End}(\mathbb{C}^{n(3g+p-3)})$
is the standard representation associated to the $n$–roots
$a_{i}^{\frac{1}{n}}$, and
$\rho^{\prime}\colon\mathcal{T}_{\tau^{\prime}}^{q}(S)\to\mathrm{End}(\mathbb{C}^{n(3g+p-3)})$
is the standard representation associated to the $n$–roots
$a_{i}^{\prime\frac{1}{n}}$, we select an isomorphism
$\Lambda\colon\mathbb{C}^{n(3g+p-3)}\to\mathbb{C}^{n(3g+p-3)}$ such that
$\rho\circ\Phi_{\tau\tau^{\prime}}^{q}(W)=\Lambda\circ\rho^{\prime}(W^{\prime})\circ\Lambda^{-1}$
for every $W^{\prime}\in\mathcal{T}_{\tau^{\prime}}^{q}(S)$ and normalized so
that $\left|\det\Lambda\right|$. This $\Lambda$ will depend on the $n$–roots
$a_{i}^{\prime\frac{1}{n}}$.

Up to diffeomorphism, there are only finitely many pairs
$(\tau,\tau^{\prime})$ where the ideal triangulation $\tau^{\prime}$ is
obtained from $\tau$ by a diagonal exchange or by an edge relabelling. This
Step 3 therefore provides finitely many linear isomorphisms
$\Lambda\colon\mathbb{C}^{n(3g+p-3)}\to\mathbb{C}^{n(3g+p-3)}$, considered as
functions of the $n$–roots $a_{i}^{\frac{1}{n}}$.

In the case of the one-puncture torus, this Step 3 corresponds to Lemma 25.

Step 4. The first three steps were preparatory, and depended only on the
surface $S$. We are now ready to tackle our general goal, with the data of an
orientation-preserving diffeomorphism $\varphi\colon S\to S$ and a
$\varphi$–invariant character
$[r]\in\mathcal{X}_{\mathrm{SL}_{2}(\mathbb{C})}(S)$ associated to a periodic
edge weight system $a^{(0)}$, $a^{(1)}$, …,
$a^{(k_{0})}\in\left(\mathbb{C}^{*}\right)^{e}$ for the ideal triangulation
sweep $\tau=\tau^{(0)}$, $\tau^{(1)}$, …, $\tau^{(k_{0}-1)}$,
$\tau^{(k_{0})}=\varphi(\tau)$, and $\varphi$–invariant puncture weights
$h_{v}$ such that

$h_{v}^{n}=a_{i_{1}}^{(0)}a_{i_{2}}^{(0)}\dots a_{i_{l}}^{(0)}$

if the puncture $v$ is adjacent to the edges $\gamma_{i_{1}}$,
$\gamma_{i_{2}}$, …, $\gamma_{i_{k}}$ of $\tau$.

With this data, start with an arbitrary choice of $n$–roots
$\left(a_{i}^{(0)}\right)^{\frac{1}{n}}$ for the edge weights of
$\tau=\tau^{(0)}$. Then, for every $k=1$, $2$, …, $k_{0}$, the rule selected
in Step 3 determines preferred $n$–roots
$\left(a_{i}^{(k)}\right)^{\frac{1}{n}}$ for the edge weights of $\tau^{(k)}$.
These $n$–roots specify a standard representation
$\rho_{k}\colon\mathcal{T}_{\tau^{(k)}}^{q}(S)\to\mathrm{End}(\mathbb{C}^{n(3g+p-3)})$
as in Step 1, and Step 3 provides a linear isomorphism
$\Lambda\colon\mathbb{C}^{n(3g+p-3)}\to\mathbb{C}^{n(3g+p-3)}$ between the
representations $\rho_{k-1}\circ\Phi^{q}_{\tau^{(k-1)}\tau^{(k)}}$ and
$\rho_{k}\colon\mathcal{T}_{\tau^{(k)}}^{q}(S)\to\mathrm{End}(\mathbb{C}^{n(3g+p-3)})$.

Since the edge weight system is periodic, each edge weight $a_{i}^{(k_{0})}$
is equal to $a_{i}^{(0)}$. However, because the $n$–roots were selected by
repeated application of Step 3, we only know that there exists integers
$m_{i}\in\mathbb{Z}$ such that
$\left(a_{i}^{(0)}\right)^{\frac{1}{n}}=q^{m_{i}}\left(a_{i}^{(k_{0})}\right)^{\frac{1}{n}}$.
Step 2 associates an isomorphism $L_{m_{1}m_{2}\dots
m_{e}}\colon\mathbb{C}^{n(3g+p-3)}\to\mathbb{C}^{n(3g+p-3)}$ to these
integers.

Finally, the argument that we used in the proof of Proposition 28 shows that
the isomorphism $\Lambda_{\varphi,r}^{q}$ associated to the data by
Proposition 4 is equal to

$\Lambda_{\varphi,r}^{q}=\Lambda_{1}\circ\Lambda_{2}\circ\dots\circ\Lambda_{k_{0}}\circ
L_{m_{1}m_{2}\dots m_{e}}$

which concludes our computation.

In its full generality, the method above involves the computation of a lot of
data. In practice, it is often possible to reduce this data by using various
combinatorial arguments.

For instance, in the case of the one-puncture torus, there exist two
(labelled) ideal triangulations up to orientation-preserving diffeomorphism,
but we were able to restrict attention to one type. Also, by combining each
diagonal exchange with an edge relabelling, we only had to consider two moves,
the “left” move and the “right” move.

A very similar simplification can be used for the four-puncture sphere, if we
restrict attention to ideal triangulations of tetrahedral type, where the
edges can be labelled as $e$, $f$, $g$, $e^{\prime}$, $f^{\prime}$,
$g^{\prime}$ in such a way that $e$, $f$, $g$ occur clockwise around a face
and $e^{\prime}$, $f^{\prime}$, $g^{\prime}$ are respectively disjoint from
$e$, $f$, $g$. (In this case, there are many more possible ideal
triangulations.) In this setup, we can arrange that the diagonal exchanges
occur along pairs of disjoint edges which, combined with suitable edge
relabelling, again leads to the consideration of only a “left” and “right”
move.

## References

  * [Ago11] Ian Agol, _Ideal triangulations of pseudo-Anosov mapping tori_ , Topology and geometry in dimension three, Contemp. Math., vol. 560, Amer. Math. Soc., Providence, RI, 2011, pp. 1–17.
  * [BB04] Stéphane Baseilhac and Riccardo Benedetti, _Quantum hyperbolic invariants of 3-manifolds with ${\rm PSL}(2,\mathbb{C})$-characters_, Topology 43 (2004), no. 6, 1373–1423.
  * [BB05] by same author, _Classical and quantum dilogarithmic invariants of flat ${\rm PSL}(2,\mathbb{C})$-bundles over 3-manifolds_, Geom. Topol. 9 (2005), 493–569.
  * [BB07] by same author, _Quantum hyperbolic geometry_ , Algebr. Geom. Topol. 7 (2007), 845–917.
  * [BFKB02] Doug Bullock, Charles Frohman, and Joanna Kania-Bartoszyńska, _The Kauffman bracket skein as an algebra of observables_ , Proc. Amer. Math. Soc. 130 (2002), no. 8, 2479–2485.
  * [BL07] Francis Bonahon and Xiaobo Liu, _Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms_ , Geom. Topol. 11 (2007), 889–937.
  * [Bon09] Francis Bonahon, _Low-dimensional geometry: from euclidean surfaces to hyperbolic knots_ , Student Mathematical Library, vol. 49, American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 2009, IAS/Park City Mathematical Subseries.
  * [BP92] Riccardo Benedetti and Carlo Petronio, _Lectures on hyperbolic geometry_ , Universitext, Springer-Verlag, Berlin, 1992.
  * [BW11a] Francis Bonahon and Helen Wong, _Kauffman brackets, character varieties and triangulations of surfaces_ , Topology and geometry in dimension three, Contemp. Math., vol. 560, Amer. Math. Soc., Providence, RI, 2011, pp. 179–194.
  * [BW11b] by same author, _Quantum traces for representations of surface groups in ${\rm SL}_{2}(\mathbb{C})$_, Geom. Topol. 15 (2011), no. 3, 1569–1615.
  * [BW16] by same author, _Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations_ , Invent. Math. 204 (2016), no. 1, 195–243.
  * [BW17] by same author, _Representations of the Kauffman bracket skein algebra II: Punctured surfaces_ , Algebr. Geom. Topol. 17 (2017), no. 6, 3399–3434.
  * [BW19] by same author, _Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality_ , Quantum Topol. 10 (2019), no. 2, 325–398.
  * [BWY22a] Francis Bonahon, Helen Wong, and Tian Yang, _Asymptotics of quantum invariants of surface diffeomorphisms II: the figure-eight knot complement_ , in preparation, 2022.
  * [BWY22b] by same author, _Asymptotics of quantum invariants of surface diffeomorphisms III: the one-puncture torus_ , in preparation, 2022.
  * [BZ05] Arkady Berenstein and Andrei Zelevinsky, _Quantum cluster algebras_ , Adv. Math. 195 (2005), no. 2, 405–455.
  * [CF00] Leonid O. Chekhov and Vladimir V. Fock, _Observables in 3D gravity and geodesic algebras_ , vol. 50, 2000, Quantum groups and integrable systems (Prague, 2000), pp. 1201–1208.
  * [FC99] Vladimir V. Fock and Leonid O. Chekhov, _Quantum Teichmüller spaces_ , Teoret. Mat. Fiz. 120 (1999), no. 3, 511–528.
  * [FG09a] Vladimir V. Fock and Alexander B. Goncharov, _Cluster ensembles, quantization and the dilogarithm_ , Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 6, 865–930.
  * [FG09b] by same author, _The quantum dilogarithm and representations of quantum cluster varieties_ , Invent. Math. 175 (2009), no. 2, 223–286.
  * [FK94] Lyudvig D. Faddeev and Rinat M. Kashaev, _Quantum dilogarithm_ , Modern Phys. Lett. A 9 (1994), no. 5, 427–434.
  * [FKBL19] Charles Frohman, Joanna Kania-Bartoszyńska, and Thang Lê, _Unicity for representations of the Kauffman bracket skein algebra_ , Invent. Math. 215 (2019), no. 2, 609–650.
  * [Fro19] Charles Frohman, Private conversation, 2019.
  * [GJS19] Iordan Ganev, David Jordan, and Pavel Safronov, _The quantum Frobenius for character varieties and multiplicative quiver varieties_ , preprint, arXiv:1901.11450.
  * [GR70] Howard Garland and M. S. Raghunathan, _Fundamental domains for lattices in (R-)rank $1$ semisimple Lie groups_, Ann. of Math. (2) 92 (1970), 279–326.
  * [Hia10] Christopher Hiatt, _Quantum traces in quantum Teichmüller theory_ , Algebr. Geom. Topol. 10 (2010), no. 3, 1245–1283.
  * [Kas94] Rinat M. Kashaev, _Quantum dilogarithm as a $6j$-symbol_, Modern Phys. Lett. A 9 (1994), no. 40, 3757–3768.
  * [Kas95] by same author, _A link invariant from quantum dilogarithm_ , Modern Phys. Lett. A 10 (1995), no. 19, 1409–1418.
  * [Kas97] by same author, _The hyperbolic volume of knots from the quantum dilogarithm_ , Lett. Math. Phys. 39 (1997), no. 3, 269–275.
  * [Lê15] Thang T. Q. Lê, _On Kauffman bracket skein modules at roots of unity_ , Algebr. Geom. Topol. 15 (2015), no. 2, 1093–1117.
  * [Liu09] Xiaobo Liu, _The quantum Teichmüller space as a noncommutative algebraic object_ , J. Knot Theory Ramifications 18 (2009), no. 5, 705–726.
  * [Liu12] by same author, _Quantum hyperbolic invariants for diffeomorphisms of small surfaces_ , Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 4, 759–770.
  * [Mar16] Bruno Martelli, _Geometric topology_ , 2016, book manuscript, available as arXiv:1610.02592.
  * [MM01] Hitoshi Murakami and Jun Murakami, _The colored Jones polynomials and the simplicial volume of a knot_ , Acta Math. 186 (2001), no. 1, 85–104.
  * [Ota96] Jean-Pierre Otal, _Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3_ , Astérisque (1996), no. 235, x+159, See also [Ota01].
  * [Ota01] by same author, _The hyperbolization theorem for fibered 3-manifolds_ , American Mathematical Society, Providence, RI, 2001, Translated from [Ota96] by Leslie D. Kay.
  * [PS00] Józef H. Przytycki and Adam S. Sikora, _On skein algebras and ${\rm Sl}_{2}({\bf C})$-character varieties_, Topology 39 (2000), no. 1, 115–148.
  * [Thu81] William P. Thurston, _The geomety and topology of $3$–manifolds_, Princeton lecture notes, available at http://library.msri.org/books/gt3m/, 1978–81.
  * [Thu88] by same author, _On the geometry and dynamics of diffeomorphisms of surfaces_ , Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431.
  * [Tur91] Vladimir G. Turaev, _Skein quantization of Poisson algebras of loops on surfaces_ , Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 6, 635–704.
  * [Wei60] André Weil, _On discrete subgroups of Lie groups_ , Ann. of Math. (2) 72 (1960), 369–384.
  * [Wei62] by same author, _On discrete subgroups of Lie groups. II_ , Ann. of Math. (2) 75 (1962), 578–602.