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# The Chen–Yang volume conjecture for knots in handlebodies

Fathi Ben Aribi and James Gosselet  UCLouvain, IRMP, Chemin du Cyclotron 2
1348 Louvain-la-Neuve
Belgium<EMAIL_ADDRESS>UCLouvain, IRMP, Chemin du Cyclotron 2
1348 Louvain-la-Neuve
Belgium<EMAIL_ADDRESS>
###### Abstract.

In 2015, Chen and Yang proposed a volume conjecture that stated that certain
Turaev–Viro invariants of an hyperbolic $3$-manifold should grow exponentially
with a rate equal to the hyperbolic volume.

Since then, this conjecture has been proven or numerically tested for several
hyperbolic $3$-manifolds, either closed or with boundary, the boundary being
either a family of tori or a family of higher genus surfaces. The current
paper now provides new numerical checks of this volume conjecture for
$3$-manifolds with one toroidal boundary component and one geodesic boundary
component.

More precisely, we study a family of hyperbolic $3$-manifolds $M_{g}$
introduced by Frigerio. Each $M_{g}$ can be seen as the complement of a knot
in an handlebody of genus $g$.

We provide an explicit code that computes the Turaev–Viro invariants of these
manifolds $M_{g}$, and we then numerically check the Chen–Yang volume
conjecture for the first six members of this family.

Furthermore, we propose an extension of the volume conjecture, where the
second coefficient of the asymptotic expansion only depends on the topology of
the boundary of the manifold. We numerically check this property for the
manifolds $M_{2},\ldots,M_{7}$ and we also observe that the second coefficient
grows linearly in the Euler characteristic $\chi(\partial M_{g})$.

###### Key words and phrases:

Turaev–Viro invariants; volume conjectures; hyperbolic volume; triangulations
of $3$-manifolds.

###### 2020 Mathematics Subject Classification:

57K16; 57K32

###### Contents

  1. 1 Introduction
  2. 2 Materials and methods
  3. 3 Preliminaries
    1. 3.1 Ideal triangulations
    2. 3.2 Hyperbolic volume
    3. 3.3 Admissible colorings
    4. 3.4 Turaev–Viro invariants
    5. 3.5 Volume conjecture
  4. 4 On Frigerio’s manifolds $M_{g}$
    1. 4.1 Frigerio’s construction
    2. 4.2 Ordered triangulations and comb representation
    3. 4.3 Hyperbolic structure
    4. 4.4 Admissible colorings
  5. 5 Annotated code
    1. 5.1 Computing the hyperbolic volume of $M_{g}$
      1. 5.1.1 The volume formula for hyperbolic tetrahedra
      2. 5.1.2 Applying the volume formula on tetrahedra of $\mathcal{T}_{g}$
    2. 5.2 Computing the Turaev–Viro invariants $TV_{r,s}(M_{g},\mathcal{T}_{g})$
    3. 5.3 Asymptotic behavior of $QV_{r,2}(M_{2})$ and $QV_{r,2}(M_{3})$
  6. 6 Numerical Results
    1. 6.1 The case of $M_{2}$
      1. 6.1.1 Triangulation
      2. 6.1.2 Hyperbolic structure
      3. 6.1.3 Admissible colorings and Turaev–Viro Invariants
      4. 6.1.4 Numerical check of the volume conjecture
    2. 6.2 The case of $M_{3}$
      1. 6.2.1 Triangulation
      2. 6.2.2 Hyperbolic structure
      3. 6.2.3 Admissible colorings and Turaev–Viro Invariants
      4. 6.2.4 Numerical check of the volume conjecture
    3. 6.3 The cases of $M_{4}$ to $M_{7}$
    4. 6.4 Behavior of the coefficient $b(M_{g})$ relative to $g$
  7. 7 Discussion and further directions

##  1\. Introduction

Quantum topology began in 1984 with the definition of the Jones Polynomial
[9]. Since then, several new invariants of knots and 3-manifolds were defined
and, inspired by quantum field theories, the _volume conjecture_ of Kashaev
[10] and its variants rose to be among the most studied conjectures in quantum
topology. What is intriguing about these volume conjectures is that they link
quantum invariants of manifolds to the hyperbolic structure of these
manifolds. In [3], Chen–Yang proposed a volume conjecture using a family
$\left\\{TV_{r,2}\right\\}_{r\in\mathbb{N}_{\geq 3}}$ of Turaev-Viro type
invariants for compact 3-manifolds.

######  Conjecture 1.1 (Conjecture 1.1, [3], Chen-Yang).

Let $M$ be a compact hyperbolic 3-manifold. Then for $r$ running over all odd
integers such that $r\geq 3$,

$\lim_{r\to\infty}\frac{2\pi}{r-2}\log\left(TV_{r,2}(M)\right)=\mathrm{Vol}(M),$

where $\mathrm{Vol}(M)$ is the hyperbolic volume of $M$.

This conjecture has since then been proven [4, 13, 16] or numerically tested
[3, 12] for several hyperbolic $3$-manifolds, mostly for manifolds without
boundary, but also manifolds with toroidal boundary and manifolds with totally
geodesic boundary of genus $g\geq 2$. However, there is currently no test of
this conjecture for an hyperbolic $3$-manifold with a boundary which has both
a toroidal component and a totally geodesic boundary component of genus $g\geq
2$. In this paper, we thus propose to test the Chen–Yang volume conjecture for
a family of $3$-manifolds $\left\\{M_{g}\right\\}_{g\in\mathbb{N}_{\geq 2}}$,
each $M_{g}$ being the exterior of a knot in a handlebody of genus $g$. These
manifolds were constructed and studied by Frigerio [5].

We will actually go one step further, as we will test an extension of
Conjecture 1.1 stated as follows:

######  Conjecture 1.2.

Let $M$ be a compact hyperbolic 3-manifold of volume $\mathrm{Vol}(M)$. Then
for $r$ running over all odd integers such that $r\geq 3$,

$TV_{r,2}(M)\underset{r\to\infty}{\sim}\omega\cdot r^{b}\cdot
e^{\frac{\mathrm{Vol}(M)}{2\pi}(r-2)}\cdot\left(1+O\left(\frac{1}{r-2}\right)\right),$

where $\omega,b\in\mathbb{R}$ are independent of $r$, or equivalently,

$\frac{2\pi}{r-2}\log\left(TV_{r,2}(M)\right)\underset{r\to\infty}{\sim}\mathrm{Vol}(M)+b\dfrac{2\pi\ln(r-2)}{r-2}+c\dfrac{1}{r-2}+O\left(\frac{1}{(r-2)^{2}}\right),$

where $b\in\mathbb{R}$ and $c\in\mathbb{C}$ are independent of $r$.

Variants of Conjecture 1.2 have been previously stated, proven and numerically
checked, notably by Chen–Yang [2, Section 6] (for some manifolds with
boundary), and by Ohtsuki [13] and Gang-Romo-Yamasaki [6] (for certain closed
manifolds, via the usual quadratic relation between Reshetikhin–Turaev
invariants and Turaev–Viro invariants). In the present paper, we aim to
numerically test Conjecture 1.2 for Frigerio’s manifolds $M_{g}$.

A yet stronger conjecture surmises that the coefficient $b$ in Conjecture 1.2
should only depend on the topology of the boundary $\partial M$. We go further
and conjecture that $b$ is linear in $\chi(\partial M)$, as stated as follows:

######  Conjecture 1.3.

Let $\mathcal{M}$ be the set of compact hyperbolic 3-manifolds for which
Conjecture 1.2 holds. Then, for all $M\in\mathcal{M}$, the coefficient $b(M)$
in Conjecture 1.2 only depends on the topology of $\partial M$, as an affine
function of the Euler characteristic $\chi(\partial M)$.

Conjecture 1.3, as stated here, might be too strong to be true. Nevertheless,
in this paper, it follows from numerical computations that Conjecture 1.3
appears to hold for Frigerio’s manifolds $M_{g}$.

After reviewing preliminaries about triangulations, hyperbolic volumes,
colorings of triangulations and Turaev–Viro invariants in Section 3, we
discuss the manifolds $M_{g}$ in Section 4: in particular we describe their
ideal triangulations $\mathcal{T}_{g}$, constructed by Frigerio. We observe
that these ideal triangulations admit an ordered structure, which means we can
orient all edges in a coherent way with face gluings and such that no triangle
admits a cycle.

######  Proposition 1.4 (Proposition 4.3).

The triangulations $\mathcal{T}_{g}$ of the manifolds $M_{g}$ admit an ordered
structure.

This property will not be used for the rest of the paper, but may be useful in
the future for studying other quantum invariants which are only defined on
ordered triangulations [1, 11], in the specific cases of these manifolds
$M_{g}$.

For two fixed integers $r\geqslant 3$ and $s\geqslant 1$, the Turaev-Viro
invariant $TV_{r,s}(M,\mathcal{T})$ of a triangulated manifold
$(M,\mathcal{T})$ is defined as a sum over a (usually large) set of admissible
colorings of the triangulation $\mathcal{T}$. Hence, in order to compute the
Turaev-Viro invariants $TV_{r,s}(M_{g},\mathcal{T}_{g})$, we first need to
describe the associated set of admissible colorings
$\mathcal{A}_{r}(M_{g},\mathcal{T}_{g})$. The main theorem of this paper
provides a description of $\mathcal{A}_{r}(M_{g},\mathcal{T}_{g})$ that is
both clearer than the original definition and easier to transform into
computer code (in Section 5.2). We now phrase it without technical details:

######  Theorem 1.5 (Theorem 4.8).

An admissible coloring of $\mathcal{T}_{g}$ has to satisfy a certain set of
admissibility conditions, which are equivalent to another explicit and more
convenient set of conditions.

The proof of Theorem 1.5 is quite lengthy but can give insights on how to
simplify admissibility conditions for other examples of triangulations. The
new set of conditions given by Theorem 1.5 will be used later on in Section
5.2.

In Section 6, we study the specific cases of $(M_{g},\mathcal{T}_{g})$ for
$2\leqslant g\leqslant 7$. We numerically compute their hyperbolic volume
(thanks to results of Frigerio [5] and Ushijima [15]) and their logarithmic
Turaev–Viro invariants
$QV_{r,2}(M_{g}):=\frac{2\pi}{r-2}\log(TV_{r,2}(M_{g},\mathcal{T}_{g}))$ for
several values of $r$. For increasing values of $r$, we observe a convergence
as expected in Conjecture 1.1 (for $2\leqslant g\leqslant 7$), and then a
surprising pattern break for $g\in\\{2,3\\}$.

######  Numerical test 1.6 (Sections 6.1.4, 6.2.4 and 6.3).

Conjectures 1.1 and 1.2 appear to hold numerically for the manifolds
$M_{2},\ldots,M_{7}$, barring possible numerical errors for $M_{2}$ and
$M_{3}$.

More precisely, the graph of the function $QV_{r,2}(M_{2})$ (resp.
$QV_{r,2}(M_{3})$) shows a converging behavior up to $r=33$ (resp. $r=31$) and
an unexpected increase after $r=33$ (resp. $r=31$). The graph of the function
$QV_{r,2}(M_{g})$ (for $4\leqslant g\leqslant 7$) shows a converging behavior.

We offer hypotheses to explain the previous pattern breaks, and we furthermore
compute an interpolating function for the data, which not only fits the pre-
break values quite well, but also provides a promising candidate for the next
term $b$ in the expected asymptotic expansion of $QV_{r,2}(M_{g})$ (see
Conjecture 1.2).

In Section 5 we provide our entire code (written in SageMath), with
annotations for clarity. In particular, we program functions that compute the
hyperbolic volumes and Turaev–Viro invariants for the manifolds $M_{g}$, and
we list several of these values in the tables of Figures 9 and 10.

Finally, we observe a linear behavior for the second coefficient $b$, as
expected in Conjecture 1.3.

######  Numerical test 1.7 (Section 6.4).

Conjecture 1.3 appears to hold numerically for the manifolds
$M_{2},\ldots,M_{7}$.

The results in this article follow in part from the Master’s thesis [7] of the
second author.

##  2\. Materials and methods

All the calculations were performed on SageMath (a free open-source
mathematics software system using Python 3), on a computer equipped with a
Intel® Core™ i5-8500 CPU @ 3.00GHz × 6 processor.

##  3\. Preliminaries

###  3.1. Ideal triangulations

In this section, we follow some conventions of [3] and [15]. A pseudo
$3$-manifold is a topological space $M$ such that each point $p$ of $M$ has an
(open) neighborhood $U_{p}$ that is homeomorphic to a cone over a surface. A
triangulation $\mathcal{T}$ of a pseudo $3$-manifold $M$ consists of a
disjoint union $T_{1}\sqcup\ldots\sqcup T_{N}$ of finitely many Euclidean
tetrahedra and of a collection of affine homeomorphisms between pairs of faces
in $T_{1}\sqcup\ldots\sqcup T_{N}$ such that the quotient space
$\left(T_{1}\sqcup\ldots\sqcup T_{N}\right)/\sim$ is homeomorphic to $M$. For
$i\in\\{0,1,2,3\\}$, we denote $\mathcal{T}^{i}$ (resp.
$\mathcal{T}^{i,\sim}$) the $i$-skeleton of the disjoint union of tetrahedra
in $\mathcal{T}$ (resp. the $i$-skeleton of the quotient space homeomorphic to
$M$). We say that an element $\nu$ of the set $\mathcal{T}^{0,\sim}$ of
vertices of $M$ is regular (resp. ideal, hyperideal) if its associated
neighborhood $U_{\nu}$ is a cone over a sphere (resp. over a torus, over a
surface of genus at least $2$). Finally, we say that a compact $3$-manifold
$N$ with boundary admits an ideal triangulation $\mathcal{T}$ if no vertices
in $\mathcal{T}^{0,\sim}$ are regular, and $N$ is obtained from the quotient
space of $\mathcal{T}$ by removing open neighborhoods $U_{\nu}$ of all
vertices $\nu$ in $\mathcal{T}^{0,\sim}$.

###  3.2. Hyperbolic volume

In [15], Ushijima gives a volume formula for a generalized hyperbolic
tetrahedron with a given angle structure. Let us detail this formula and its
components. Let $T=T(A,B,C,D,E,F)$ be a generalized tetrahedron in the
hyperbolic space $\mathbb{H}^{3}$ whose angle structure is as in Figure 1
($A,\ldots,F\in[0,\pi]$ are dihedral angles).

$v_{0}$$v_{1}$$v_{2}$$v_{3}$CEDABF Figure 1. The configuration of dihedral
angles on $T$.

Let $G$ denote the associated Gram matrix:

$G=\left(\begin{array}[]{cccc}1&-\cos\,A&-\cos\,B&-\cos\,F\\\
-\cos\,A&1&-\cos\,C&-\cos\,E\\\ -\cos\,B&-\cos\,C&1&-\cos\,D\\\
-\cos\,F&-\cos\,E&-\cos\,D&1\end{array}\right).$

Let $\mathrm{Li_{2}}(z)$ be the dilogarithm function defined for
$z\in\mathbb{C}\setminus[1,\infty)$ in [15, Introduction] by the analytic
continuation of the following integral:

$\mathrm{Li_{2}}(x):=-\int_{0}^{x}\frac{\log(1-t)}{t}\text{d}t\quad\text{for
}x\in\mathbb{R}_{>0}.$

Let $I$ be the principal square root of $-1$ , $a:=\exp(I\cdot A)$,
$b:=\exp(I\cdot B)$, …, $f:=\exp(I\cdot F)$ and let $U(z,T)$ be the complex
valued function defined as follows:

$\displaystyle U(z,T):=\frac{1}{2}($
$\displaystyle\mathrm{Li_{2}}(z)+\mathrm{Li_{2}}(abdez)+\mathrm{Li_{2}}(acdfz)+\mathrm{Li_{2}}(bcefz)$
$\displaystyle-\mathrm{Li_{2}}(-abcz)-\mathrm{Li_{2}}(-aefz)-\mathrm{Li_{2}}(-bdfz)-\mathrm{Li_{2}}(-cdez)).$

We denote by $z_{+}$ and $z_{-}$ the two complex numbers defined as follows:

$\displaystyle z_{\pm}:=$
$\displaystyle-2\frac{\sin\,A\sin\,D+\sin\,B\sin\,E+\sin\,C\sin\,F\pm\sqrt{\text{det}\,G}}{ad+be+cf+abf+ace+bcd+def+abcdef},$

where $\sqrt{\text{det}\,G}\in I\mathbb{R}_{>0}$ is the principal square root
of $\text{det}\,G$.

######  Proposition 3.1 (Ushijima, [15] Theorem 1.1).

The hyperbolic volume $\mathrm{Vol}(T)$ of a generalized tetrahedron
$T=T(A,B,C,D,E,F)$ is given as follows:

$\mathrm{Vol}(T)=\frac{1}{2}\mathcal{I}(U(z_{-},T)-U(z_{+},T)),$

where $\mathcal{I}$ means the imaginary part.

In Section 5.1.1 we provide a code to compute the previous functions and
formula.

###  3.3. Admissible colorings

We will mostly follow the notations, conventions and definitions of [3,
Section 2]. For the remainder of this paper, let us fix a pair
$(r,s)\in\mathbb{N}^{2}$ such that $r\geq 3$ and $s\geq 1$. We will only
specify $s=2$ when studying the Chen–Yang volume conjecture.

######  Notation 3.2.

Let $\frac{\mathbb{N}}{2}=\left\\{0,\frac{1}{2},1,\frac{3}{2},...\right\\}$
denote the set of non-negative half-integers. Let
$\frac{\mathbb{N}_{odd}}{2}=\left\\{\frac{1}{2},\frac{3}{2},\frac{5}{2},...\right\\}$
denote the set of non-negative half-odd-integers. Let $I_{r}$ denote the
subset $\left\\{0,\frac{1}{2},1,...,\frac{r-2}{2}\right\\}$ of
$\frac{\mathbb{N}}{2}$.

As a convention, we let $\sqrt{-x}=I\sqrt{x}$ for $x\geq 0$.

######  Definition 3.3.

A triple $(i,j,k)$ of elements of $I_{r}$ is called _admissible_ if it
satisfies the following conditions:

  1. (i)

    1. (a)

$i+j\geq k$,

    2. (b)

$j+k\geq i$,

    3. (c)

$k+i\geq j$,

  2. (ii)

$i+j+k\in\mathbb{N}$,

  3. (iii)

$i+j+k\leq r\scalebox{1.8}[1.0]{-}2$.

######  Definition 3.4.

Let $M$ be an hyperbolic compact 3-manifold with boundary that admits an ideal
triangulation $\mathcal{T}$. A _coloring at level r of $(M,\mathcal{T})$_ is
an application $c:\mathcal{T}^{1,\sim}\to I_{r}$. The coloring is called
_admissible_ if for every $T\in\mathcal{T}^{3}$, the triples
$\left(c([e_{01}]),c([e_{02}]),c([e_{03}])\right)$,
$\left(c([e_{01}]),c([e_{12}]),c([e_{13}])\right)$,
$\left(c([e_{02}]),c([e_{12}]),c([e_{23}])\right)$ and
$\left(c([e_{03}]),c([e_{13}]),c([e_{23}])\right)$ are admissible, where
$[e_{kl}]\in\mathcal{T}^{1,\sim}$ denotes the equivalence class under $\sim$
of the edge $e_{kl}$ of $T$ (for $k,l\in\\{0,1,2,3\\}$ such that $k\neq l$).

###  3.4. Turaev–Viro invariants

Recall that $r,s\in\mathbb{N}$ are such that $r\geq 3$ and $s\geq 1$.

For $n\in\mathbb{N}$, the _quantum number_ $[n]$ is the real number defined by

$[n]:=\frac{\sin\left(\frac{ns\pi}{r}\right)}{\sin\left(\frac{s\pi}{r}\right)}\in\mathbb{R}.$

For $n\in\mathbb{N}$, the _quantum factorial_ $[n]!$ is defined by

$[n]!:=[n][n\scalebox{1.8}[1.0]{-}1]...[2][1]\in\mathbb{R},$

and, as a convention, $[0]!=1$. For an admissible triple
$(i,j,k)\in(I_{r})^{3}$, we define

$\Delta(i,j,k):=\sqrt{\frac{[i+j\scalebox{1.8}[1.0]{-}k]![i\scalebox{1.8}[1.0]{-}j+k]![\scalebox{1.8}[1.0]{-}i+j+k]!}{[i+j+k+1]!}}.$

######  Definition 3.5 (Quantum $6j$-symbols).

Let $(i,j,k,l,m,n)$ be a 6-tuple of elements of $I_{r}$ such that $(i,j,k)$,
$(j,l,n)$, $(i,m,n)$ and $(k,l,m)$ are admissible.

Let $T_{1}=i+j+k$, $T_{2}=j+l+n$, $T_{3}=i+m+n$, $T_{4}=k+l+m$,
$Q_{1}=i+j+l+m$, $Q_{2}=i+k+l+n$ and $Q_{3}=j+k+m+n$.

Then _the quantum $6j$-symbol_ for the 6-tuple $(i,j,k,l,m,n)$ is defined by

$\displaystyle\begin{vmatrix}i&j&k\\\ l&m&n\end{vmatrix}:=$
$\displaystyle\sqrt{\scalebox{1.8}[1.0]{-}1}^{-2(i+j+k+l+m+n)}\Delta(i,j,k)\Delta(j,l,n)\Delta(i,m,n)\Delta(k,l,m)$
$\displaystyle\cdot$
$\displaystyle\sum_{z=\max\\{T_{1},T_{2},T_{3},T_{4}\\}}^{\min\\{Q_{1},Q_{2},Q_{3}\\}}\frac{(\scalebox{1.8}[1.0]{-}1)^{z}[z+1]!}{[z\scalebox{1.8}[1.0]{-}T_{1}]![z\scalebox{1.8}[1.0]{-}T_{2}]![z\scalebox{1.8}[1.0]{-}T_{3}]![z\scalebox{1.8}[1.0]{-}T_{4}]![Q_{1}\scalebox{1.8}[1.0]{-}z]![Q_{2}\scalebox{1.8}[1.0]{-}z]![Q_{3}\scalebox{1.8}[1.0]{-}z]!}.$

######  Proposition 3.6 (Allowed symbol permutations).

Let $(i,j,k,l,m,n)$ be a 6-tuple of elements of $I_{r}$ such that $(i,j,k)$,
$(j,l,n)$, $(i,m,n)$, $(k,l,m)$ are admissible.

Then, we have the following allowed permutations:

$\begin{vmatrix}i&j&k\\\ l&m&n\end{vmatrix}=\begin{vmatrix}j&i&k\\\
m&l&n\end{vmatrix}=\begin{vmatrix}i&k&j\\\
l&n&m\end{vmatrix}=\begin{vmatrix}i&m&n\\\
l&j&k\end{vmatrix}=\begin{vmatrix}l&m&k\\\
i&j&n\end{vmatrix}=\begin{vmatrix}l&j&n\\\ i&m&k\end{vmatrix}.$

Let $M$ be a pseudo-$3$-manifold that admits an triangulation $\mathcal{T}$.
Let $R\subset\mathcal{T}^{0,\sim}$ denote the set of regular vertices. We
define the _regular vertices term_ as $N:=\left(\sum_{i\in
I_{r}}w_{i}^{2}\right)^{-|R|}$. Let $c:\mathcal{T}^{1,\sim}\to I_{r}$ be an
admissible coloring at level $r$ of $(M,\mathcal{T})$. For
$\eta\in\mathcal{T}^{1,\sim}$, we define the _edge term_
$|\eta|_{c}:=w_{c(\eta)}$, where $w_{i}:=(\scalebox{1.8}[1.0]{-}1)^{2i}[2i+1]$
for $i\in I_{r}$. For $T\in\mathcal{T}^{3}$, we define the _tetrahedron term_
as

$|T|_{c}=\begin{vmatrix}c([e_{01}])&c([e_{02}])&c([e_{12}])\\\
c([e_{23}])&c([e_{13}])&c([e_{03}])\end{vmatrix},$

where $[e_{kl}]\in\mathcal{T}^{1,\sim}$ denotes the equivalence class under
$\sim$ of the edge $e_{kl}$ of $T$ (for $k,l\in\\{0,1,2,3\\}$ such that $k\neq
l$).

######  Definition 3.7 (Turaev–Viro invariant).

Let $m=\left|\mathcal{T}^{1,\sim}\right|$. Let
$\mathcal{A}_{r}(M,\mathcal{T}):=$

$\left\\{(c(\eta_{0}),...,c(\eta_{m}))\ |\ c:\mathcal{T}^{1,\sim}\to
I_{r}\text{ is an admissible coloring at level }r\text{ of
}(M,\mathcal{T})\right\\}.$

We define the _Turaev–Viro invariant_ of $M$ as

$TV_{r,s}(M,\mathcal{T}):=N\sum_{(c(\eta_{0}),...,c(\eta_{m}))\in\mathcal{A}_{r}(M,\mathcal{T})}\prod_{i=0}^{m}|\eta_{i}|_{c}\prod_{T\in
X^{3}}|T|_{c}.$

In [3, Theorem 2.6], Chen and Yang prove that the previously defined
Turaev–Viro invariant of $M$ does not depend on the triangulation
$\mathcal{T}$.

###  3.5. Volume conjecture

We can now state the Chen–Yang volume conjecture:

######  Conjecture 3.8 ([3], Conjecture 1.1).

Let $M$ be a compact hyperbolic 3-manifold. Then for $r$ running over all odd
integers such that $r\geq 3$,

$\lim_{r\to\infty}QV_{r,2}(M)=\lim_{r\to\infty}\frac{2\pi}{r-2}\log\left(TV_{r,2}(M)\right)=\mathrm{Vol}(M),$

where $QV_{r,2}(M):=\frac{2\pi}{r-2}\log\left(TV_{r,2}(M)\right)$ and
$\mathrm{Vol}(M)$ is the hyperbolic volume of $M$.

######  Remark 3.9.

In [2, Section 6.1], Chen and Yang discuss the potential asymptotic behavior
of $QV_{r,2}(M)$ in more detail than in Conjecture 3.8. In particular,
observations for specific $M$ with one boundary component lead them to ask if

$QV_{r,2}(M)=\mathrm{Vol}(M)+b\dfrac{\ln(r-2)}{r-2}+c\dfrac{1}{r-2}+O\left(\dfrac{1}{(r-2)^{2}}\right),$

where the numbers $b,c$ would depend only on $M$. In the examples they study,
they find $b$ to be close to $\pi$ when $M$ has one toroidal boundary
component and $-3\pi$ when $M$ has one geodesic boundary component of genus
$2$. We rephrased this general question as Conjecture 1.2.

We will study in Sections 6.1.4 and 5.3 how likely these asymptotic behaviors
seem for the manifolds $M_{2},M_{3},\ldots,M_{7}$ (which have two boundary
components each, of different genera).

##  4\. On Frigerio’s manifolds $M_{g}$

###  4.1. Frigerio’s construction

This section follows closely the construction in [5]. Let
$g\in\mathbb{N}_{\geq 2}$. Let $S^{3}$ be the one point compactification of
$\mathbb{R}^{3}$.

Figure 2. $\Gamma_{g}$ has two components: $\Gamma^{0}_{g}$ is a knot and
$\Gamma^{1}_{g}$ is a graph with $g+1$ edges and two vertices. Source: [5,
Figure 1] Figure 3. The graphs $\Gamma_{2}$ (on the left) and $\Gamma_{3}$ (on
the right). Source: [5, Figure 2]

Let $\Gamma_{g}\subset S^{3}$ be the graph shown in Figure 2 (see Figure 3 for
the cases $g=2$ and $g=3$). Let us denote by $\Gamma^{0}_{g}$ and
$\Gamma^{1}_{g}$ the two connected components of $\Gamma_{g}$, where
$\Gamma^{0}_{g}$ is a knot and $\Gamma^{1}_{g}$ has two vertices and $g+1$
edges. Let $U(\Gamma^{0}_{g})$ and $U(\Gamma^{1}_{g})$ denote open regular
neighbourhoods of $\Gamma^{0}_{g}$ and $\Gamma^{1}_{g}$. Then $M_{g}$ is
defined as the compact 3-manifold
$M_{g}:=S^{3}\setminus\left\\{U(\Gamma^{0}_{g}),U(\Gamma^{1}_{g})\right\\}$
with boundary $\partial M=(\partial M)_{0}\cup(\partial M)_{1}$ such that
$(\partial M)_{0}=\partial U(\Gamma^{0}_{g})\cong S^{1}\times S^{1}$ and
$(\partial M)_{1}=\partial U(\Gamma^{1}_{g})\cong\Sigma_{g}$ a genus $g$
surface.

######  Remark 4.1.

The 3-manifold $M_{g}$ is the exterior of a knot in the handlebody of genus
$g$ as seen in Figure 4.

Figure 4. The 3-manifold $M_{2}$ as a knot complement in the handlebody of
genus 2. Source: [5, Figure 3]

In [5, Section 2], Frigerio constructs an ideal triangulation of $M_{g}$. This
construction is illustrated for the cases $g=2$ and $g=3$ in Figure 5. Let
$P_{g}$ be the double cone with apices $ap_{1}$ and $ap_{2}$ and based on the
regular ($2g+2$)-gon whose vertices are $p_{0}$, $p_{1}$, …, $p_{2g+1}$. Let
$\mathring{P}_{g}$ be $P_{g}$ with its vertices removed. Let $Y_{g}$ be the
topological space obtained by gluing the faces of $\mathring{P}_{g}$ according
to the following rules:

  * •

For any $i=0,2,...,2g$ , the face $[ap_{1},p_{i},p_{i+1}]$ is identified with
the face $[p_{i+1},p_{i+2},v_{2}]$ (with $v_{1}$ identified with $p_{i+1}$,
$p_{i}$ identified with $p_{i+2}$ and $p_{i+1}$ identified with $ap_{2}$),

  * •

For any $i=1,3,...,2g+1$ , the face $[ap_{1},p_{i},p_{i+1}]$ is identified
with the face $[p_{i+2},v_{2},p_{i+1}]$ (with $ap_{1}$ identified with
$p_{i+2}$, $p_{i}$ identified with $ap_{2}$ and $p_{i+1}$ identified with
$p_{i+1}$).

######  Proposition 4.2 (Proposition 2.1, [5]).

For any $g\geq 2$, $Y_{g}$ is homeomorphic to the interior of $M_{g}$.

We can then subdivide $P_{g}$ into $2g+2$ tetrahedra by adding the vertical
edge between the two apices $ap_{1}$ and $ap_{2}$ (in red in Figure 5). Such
tetrahedra give an ideal triangulation of $M_{g}$.

$ap_{1}$$ap_{2}$$p_{0}$$p_{1}$$p_{2}$$p_{3}$$p_{4}$$p_{5}$

$ap_{1}$$ap_{2}$$p_{0}$$p_{1}$$p_{2}$$p_{3}$$p_{4}$$p_{5}$$p_{6}$$p_{7}$

Figure 5. Representation of $Y_{2}$ (left) and $Y_{3}$ (right) with the red
vertical edge between the two apices and identified edges and vertices have
the same color.

###  4.2. Ordered triangulations and comb representation

We refer to [1, 11] for the following definitions. A triangulation is called
ordered when it is endowed with an order on each quadruplet of vertices of
each tetrahedron, such that the face gluings respect the vertex order. An
equivalent property is that we can orient the edges of the triangulation in a
compatible way with the face gluings and such that there are no cycles of
length $3$.

$0$$2$$3$$1$$v_{0}$$v_{2}$$v_{1}$$v_{3}$ Figure 6. An ordered tetrahedron $T$
(right) and its associated comb $C(T)$ (left).

A comb $C$ is a line together with four spikes pointing in the same direction.
The spikes are numbered 0, 1, 2 and 3 going from left to right, with the
spikes pointing upward. The comb representation of an ordered triangulation
consists in associating a comb $C(T)$ to each tetrahedron $T$ of the
triangulation, as in Figure 6; each spike numbered $i$ corresponds to a face
$f^{i}$ opposed to a vertex $v_{i}$ for $i\in\\{0,1,2,3\\}$, and we join by a
line the spike $i$ of $C(T)$ to the spike $j$ of $C(T^{\prime})$ if the $i$-th
face of $T$ is glued to the $j$-th face of $T^{\prime}$.

The comb representation provides a compact way of representing an ordered
triangulation, while containing all the information of this triangulation.
Moreover, comb representations are convenient for studying certain quantum
invariants of ordered triangulations (such as the Teichmüller TQFT [1]). In
this sense, it is interesting to observe when a given triangulation admits an
ordered structure. This is the case for Firgerio’s triangulations
$\mathcal{T}_{g}$, as we now state:

######  Proposition 4.3.

The triangulations $\mathcal{T}_{g}$ of the manifolds $M_{g}$ admit an ordered
structure.

The proof is quite standard, in that we exhibit a specific ordered structure
(more details are in [7]). For brevity we will only refer to the examples of
$g=2$ and $g=3$ represented in Figures 12 and 15. The associated comb
representation for $g=2$ is drawn on Figure 7.

$0$$1$$2$$3$$0$$1$$2$$3$$0$$1$$2$$3$$0$$1$$2$$3$$0$$1$$2$$3$$0$$1$$2$$3$
Figure 7. Comb representation of $\mathcal{T}_{2}$.

###  4.3. Hyperbolic structure

In [5, Section 2], Frigerio provides the unique angle structure on the
tetrahedra of the ideal triangulation $\mathcal{T}_{g}$ that corresponds to
the unique complete hyperbolic structure on the manifold $M_{g}$. Setting
$\alpha_{g}=\frac{\pi}{2g+2}$ , $\beta_{g}=2\alpha_{g}$,
$\gamma_{g}=\arccos((2\cos\alpha_{g})^{-1})$ and $\delta_{g}=\pi-2\gamma_{g}$,
the complete angle structure on $\mathcal{T}_{g}$ follows the pattern of
Figure 8: half of the $2g+2$ tetrahedra are as the one on the left, and the
other half are as the one on the right.

$v_{0}$$v_{1}$$v_{2}$$v_{3}$$\beta_{g}$$\gamma_{g}$$\alpha_{g}$$\gamma_{g}$$\alpha_{g}$$\delta_{g}$

$v_{0}$$v_{1}$$v_{2}$$v_{3}$$\beta_{g}$$\alpha_{g}$$\gamma_{g}$$\alpha_{g}$$\gamma_{g}$$\delta_{g}$

Figure 8. Frigerio’s angle structure on $\mathcal{T}_{g}$

Both tetrahedra of Figure 8 have the same volume, and the volume of $M_{g}$ is
exactly $2g+2$ times higher. Values of these volumes are computed with
SageMath (see Section 5.1.2) and listed in Figure 9.

###  4.4. Admissible colorings

In this section we study the set of admissible colorings for the
triangulations $\mathcal{T}_{g}$. The following three lemmas follow from
standard arguments, are not restricted to the manifolds $M_{g}$, and will be
used in the proof of Theorem 4.8. See [7] for details.

######  Lemma 4.4.

Let $i,k\in I_{r}$. Then $(i,i,k)$ is admissible if and only if
$k\in\mathbb{N}$ and $\frac{k}{2}\leq i\leq\frac{r-2-k}{2}$.

######  Lemma 4.5.

Let $i,j,k,l,m\in I_{r}$ such that $(i,j,k)$ and $(k,l,m)$ are admissible,
then $i+j+l+m\in\mathbb{N}$.

######  Lemma 4.6.

Let $i,j,k\in I_{r}$ such that $k\in\mathbb{N}$. Then the following conditions
are equivalent:

  1. (i)

$i+j+k\in\mathbb{N}$,

  2. (ii)

$i+j\in\mathbb{N}$,

  3. (iii)

either $i,j\in\mathbb{N}$ or $i,j\in\frac{\mathbb{N}_{odd}}{2}$.

######  Remark 4.7.

Lemma 4.6 can be applied to condition (ii) of Definition 3.3 when $i$, $j$ or
$k$ is in $\mathbb{N}$.

Let $g\in\mathbb{N}_{\geq 2}$. For a general coloring $c$ at level $r$ of
$(M_{g},\mathcal{T}_{g})$, we need the following triples of elements of
$I_{r}$ to be admissible:

$(a,b,b)$, |  $(a,c_{g},c_{0})$, |  $(b,c_{g},c_{g})$, |  $(b,c_{g},c_{0})$,
---|---|---|---
and for all $i\in\left\\{0,1,2,...,g-1\right\\}$, |  $(a,c_{i},c_{i+1})$, |  $(b,c_{i},c_{i})$, |  $(b,c_{i},c_{i+1})$.

These admissibility conditions can be rewritten like in the next theorem.

######  Theorem 4.8.

Let $a$, $b$, $c_{0}$, $c_{1}$, $c_{2}$, …, $c_{g}\in I_{r}$.

The conditions $(A)$, $(B)$, $(C)$, $(D)$, $(E)$, $(F)$ and $(G)$ defined
below are simultaneously satisfied if and only if the conditions $(1)$, $(2)$,
$(3)$, $(4)$, $(5)$ and $(6)$ defined below are simultaneously satisfied.

The conditions $(A)$, $(B)$, $(C)$, $(D)$, $(E)$, $(F)$ and $(G)$ are written:

  1. (A)

$(a,b,b)$ is admissible,

  2. (B)

$(a,c_{g},c_{0})$ is admissible,

  3. (C)

$(b,c_{g},c_{g})$ is admissible,

  4. (D)

$(b,c_{g},c_{0})$ is admissible,

  5. (E)

$\forall i\in\left\\{0,1,2,...,g-1\right\\}$, $(a,c_{i},c_{i+1})$ is
admissible,

  6. (F)

$\forall i\in\left\\{0,1,2,...,g-1\right\\}$, $(b,c_{i},c_{i})$ is admissible,

  7. (G)

$\forall i\in\left\\{0,1,2,...,g-1\right\\}$, $(b,c_{i},c_{i+1})$ is
admissible,

The conditions $(1)$, $(2)$, $(3)$, $(4)$, $(5)$ and $(6)$ are written:

  1. (1)

$a\in\mathbb{N}$,

  2. (2)

$b\in\mathbb{N}$,

  3. (3)

either $\forall i\in\left\\{0,1,2...,g\right\\}$, $c_{i}\in\mathbb{N}$ or
$\forall i\in\left\\{0,1,2,...,g\right\\}$,
$c_{i}\in\frac{\mathbb{N}_{odd}}{2}$,

  4. (4)

$\frac{a}{2}\leq b\leq\frac{r-2-a}{2}$,

  5. (5)

    1. (a)

$\forall i\in\left\\{0,1,2...,g-1\right\\}$, $\frac{b}{2}\leq
c_{i+1}\leq\frac{r-2-b}{2}$,

    2. (b)

$\forall i\in\left\\{0,1,2...,g-1\right\\}$, $a-c_{i}\leq c_{i+1}\leq
r-2-a-c_{i}$,

    3. (c)

$\forall i\in\left\\{0,1,2...,g-1\right\\}$, $c_{i}-a\leq c_{i+1}\leq
a+c_{i}$,

    4. (d)

$\forall i\in\left\\{0,1,2...,g-1\right\\}$, $c_{i}-b\leq c_{i+1}\leq
b+c_{i}$,

  6. (6)

    1. (a)

$\frac{b}{2}\leq c_{0}\leq\frac{r-2-b}{2}$,

    2. (b)

$a-c_{g}\leq c_{0}\leq r-2-a-c_{g}$,

    3. (c)

$c_{g}-a\leq c_{0}\leq a+c_{g}$,

    4. (d)

$c_{g}-b\leq c_{0}\leq b+c_{g}$.

###### Proof.

Let $a$, $b$, $c_{0}$, $c_{1}$, $c_{2}$, …, $c_{g}\in I_{r}$.

Step 1 : $(\ref{conditionA})\iff(\ref{condition1})\wedge(\ref{condition4})$;

This step follows immediately from Lemma 4.4.

Step 2 :
$(\ref{conditionA})\wedge(\ref{conditionE})\iff(\ref{condition1})\wedge(\ref{condition3})\wedge(\ref{condition4})\wedge(\ref{condition5b})\wedge(\ref{condition5c})$;

By Definition 3.3, Condition $(\ref{conditionE})$ is equivalent to the
following conditions :

  1. (Ei)

    1. (a)

$\forall i\in\left\\{0,1,2...,g-1\right\\}$, $c_{i}+c_{i+1}\geq a$,

    2. (b)

$\forall i\in\left\\{0,1,2...,g-1\right\\}$, $c_{i}+a\geq c_{i+1}$,

    3. (c)

$\forall i\in\left\\{0,1,2...,g-1\right\\}$, $a+c_{i+1}\geq c_{i}$,

  2. (Eii)

$\forall i\in\left\\{0,1,2...,g-1\right\\}$, $c_{i}+c_{i+1}+a\in\mathbb{N}$,

  3. (Eiii)

$\forall i\in\left\\{0,1,2...,g-1\right\\}$, $c_{i}+c_{i+1}+a\leq
r\scalebox{1.8}[1.0]{-}2$.

To prove Step 2, we first remark that:

$(E\ref{ia})\wedge(E\ref{iii})\iff(\ref{condition5b}),$
$(E\ref{ib})\wedge(E\ref{ic})\iff(\ref{condition5c}).$

It remains to prove that :

$(\ref{conditionA})\wedge(E\ref{ii})\iff(\ref{condition1})\wedge(\ref{condition3})\wedge(\ref{condition4}).$

It follows from Step 1 that :

$(\ref{conditionA})\wedge(E\ref{ii})\iff(\ref{condition1})\wedge(\ref{condition4})\wedge(E\ref{ii}).$

From Lemma 4.6, it follows that

$(\ref{condition1})\wedge(\ref{condition4})\wedge(E\ref{ii})\iff(\ref{condition1})\wedge(\ref{condition4})\wedge\left(\forall
i\in\left\\{0,1,2...,g-1\right\\},\left(c_{i},c_{i+1}\in\mathbb{N}\right)\lor\left(c_{i},c_{i+1}\in\frac{\mathbb{N}_{odd}}{2}\right)\right).$

Finally, it follows by a quick induction that:

$(\ref{condition1})\wedge(\ref{condition4})\wedge\left(\forall
i\in\left\\{0,1,2...,g-1\right\\},\left(c_{i},c_{i+1}\in\mathbb{N}\right)\lor\left(c_{i},c_{i+1}\in\frac{\mathbb{N}_{odd}}{2}\right)\right)\iff(\ref{condition1})\wedge(\ref{condition4})\wedge(\ref{condition3}).$

Step 3 :
$(\ref{conditionA})\wedge(\ref{conditionB})\wedge(\ref{conditionE})\iff(\ref{condition1})\wedge(\ref{condition3})\wedge(\ref{condition4})\wedge(\ref{condition5b})\wedge(\ref{condition5c})\wedge(\ref{condition6b})\wedge(\ref{condition6c})$;

From Step 2, it follows that:

$(\ref{conditionA})\wedge(\ref{conditionE})\iff(\ref{condition1})\wedge(\ref{condition3})\wedge(\ref{condition4})\wedge(\ref{condition5b})\wedge(\ref{condition5c}).$

By Definition 3.3, Condition $(\ref{conditionB})$ is equivalent to the
following conditions :

  1. (Bi)

    1. (a)

$c_{g}+c_{0}\geq a$,

    2. (b)

$c_{g}+a\geq c_{0}$,

    3. (c)

$a+c_{0}\geq c_{g}$,

  2. (Bii)

$c_{g}+c_{0}+a\in\mathbb{N}$,

  3. (Biii)

$c_{g}+c_{0}+a\leq r\scalebox{1.8}[1.0]{-}2$.

To prove Step 3, we first remark that:

$(B\ref{iaB})\wedge(B\ref{iiiB})\iff(\ref{condition6b}),$
$(B\ref{ibB})\wedge(B\ref{icB})\iff(\ref{condition6c}).$

Finally, let us prove the following implication which will imply Step 3:

$(\ref{conditionA})\wedge(\ref{conditionE})\implies(B\ref{iiB}).$

From Step 2, it follows that

$(\ref{conditionA})\wedge(\ref{conditionE})\implies(\ref{condition1})\wedge(\ref{condition3}).$

Furthermore, using Lemma 4.6, we have that:

$(\ref{condition1})\wedge(\ref{condition3})\implies(B\ref{iiB}).$

Step 4 :
$(\ref{conditionC})\wedge(\ref{conditionF})\iff(\ref{condition2})\wedge(\ref{condition5a})\wedge(\ref{condition6a})$;

This step follows immediately from Lemma 4.4.

Step 5 :
$(\ref{conditionA})\wedge(\ref{conditionB})\wedge(\ref{conditionC})\wedge(\ref{conditionE})\wedge(\ref{conditionF})\wedge(\ref{conditionG})\iff$
$(\ref{condition1})\wedge(\ref{condition2})\wedge(\ref{condition3})\wedge(\ref{condition4})\wedge(\ref{condition5a})\wedge(\ref{condition5b})\wedge(\ref{condition5c})\wedge(\ref{condition5d})\wedge(\ref{condition6a})\wedge(\ref{condition6b})\wedge(\ref{condition6c})$;

From Step 3, it follows that:

$(\ref{conditionA})\wedge(\ref{conditionB})\wedge(\ref{conditionE})\iff(\ref{condition1})\wedge(\ref{condition3})\wedge(\ref{condition4})\wedge(\ref{condition5b})\wedge(\ref{condition5c})\wedge(\ref{condition6b})\wedge(\ref{condition6c}).$

From Step 4, it follows that:

$(\ref{conditionC})\wedge(\ref{conditionF})\iff(\ref{condition2})\wedge(\ref{condition5a})\wedge(\ref{condition6a}).$

By Definition 3.3, Condition $(\ref{conditionG})$ is equivalent to the
following conditions :

  1. (Gi)

    1. (a)

$\forall i\in\left\\{0,1,2...,g-1\right\\}$, $c_{i}+c_{i+1}\geq b$,

    2. (b)

$\forall i\in\left\\{0,1,2...,g-1\right\\}$, $c_{i}+b\geq c_{i+1}$,

    3. (c)

$\forall i\in\left\\{0,1,2...,g-1\right\\}$, $b+c_{i+1}\geq c_{i}$,

  2. (Gii)

$\forall i\in\left\\{0,1,2...,g-1\right\\}$, $c_{i}+c_{i+1}+b\in\mathbb{N}$,

  3. (Giii)

$\forall i\in\left\\{0,1,2...,g-1\right\\}$, $c_{i}+c_{i+1}+b\leq
r\scalebox{1.8}[1.0]{-}2$.

Following what precedes, in order to prove Step 5, we only need to prove that:

  1. Step 5.1

$(\ref{conditionC})\wedge(\ref{conditionF})\implies(G\ref{iab})\wedge(G\ref{iiib})$,

  2. Step 5.2

$(G\ref{ibb})\wedge(G\ref{icb})\iff(\ref{condition5d})$,

  3. Step 5.3

$(\ref{conditionA})\wedge(\ref{conditionE})\implies(G\ref{iib})$.

Let us prove Step 5.1. From Step 4, it follows that

$(\ref{conditionC})\wedge(\ref{conditionF})\implies(\ref{condition5a})\wedge(\ref{condition6a}).$

Furthermore, we have by adding inequalities that :

$(\ref{condition5a})\wedge(\ref{condition6a})\implies(G\ref{iab})\wedge(G\ref{iiib}).$

Step 5.2 follows immediately from the definitions.

Finally, let us prove Step 5.3. From Step 2, it follows that

$(\ref{conditionA})\wedge(\ref{conditionE})\implies(\ref{condition1})\wedge(\ref{condition3}).$

Furthermore, using Lemma 4.6, we have that:

$(\ref{condition1})\wedge(\ref{condition3})\implies(G\ref{iib}).$

This concludes the proof of Step 5.

Step 6 :
$(\ref{conditionA})\wedge(\ref{conditionB})\wedge(\ref{conditionC})\wedge(\ref{conditionD})\wedge(\ref{conditionE})\wedge(\ref{conditionF})\wedge(\ref{conditionG})\iff$
$(\ref{condition1})\wedge(\ref{condition2})\wedge(\ref{condition3})\wedge(\ref{condition4})\wedge(\ref{condition5a})\wedge(\ref{condition5b})\wedge(\ref{condition5c})\wedge(\ref{condition5d})\wedge(\ref{condition6a})\wedge(\ref{condition6b})\wedge(\ref{condition6c})\wedge(\ref{condition6d})$;

We prove Step 6 almost exactly as we proved Step 5. The only difference being
the $(G)$ conditions become $(D)$ conditions and $(5d)$ becomes $(6d)$.

This concludes the proof of the theorem. ∎

As we will see, Theorem 4.8 can be used to simplify the formula and
computation of the Turaev–Viro invariants for the manifolds $M_{g}$, and to
write a code for computing these invariants numerically (see Section 5.2).

##  5\. Annotated code

The following codes have been written with SageMath, a free open-source
mathematics software system using Python 3. In this section, let us fix a pair
$(r,s)\in\mathbb{N}^{2}$ such that $r\geq 3$ and $s\geq 1$. Let
$g\in\mathbb{N}_{\geqslant 2}$. Let us recall the following notation:

######  Notation 5.1.

Let $\frac{\mathbb{N}}{2}=\\{0,\frac{1}{2},1,...\\}$ denote the set of non-
negative half integers. Let
$\frac{\mathbb{N}_{odd}}{2}=\\{\frac{1}{2},\frac{3}{2},...\\}$ denote the set
of non-negative half-odd-integers. Let $I_{r}$ denote the subset
$\left\\{0,\frac{1}{2},1,...,\frac{r-2}{2}\right\\}$ of
$\frac{\mathbb{N}}{2}$.

###  5.1. Computing the hyperbolic volume of $M_{g}$

In this section, we construct a function which computes the volume of a
tetrahedron from its dihedral angles, via Ushijima’s volume formula (see
Proposition 3.1). We then apply this function to the manifolds $M_{g}$ with
triangulations $\mathcal{T}_{g}$.

####  5.1.1. The volume formula for hyperbolic tetrahedra

We follow the steps and notation of Section 3.2. We first import the NumPy
package which is an easy to use, open source package in Python. This package
contains the function pi giving an approximate value of $\pi$ that we will use
later on. Functions imported from this package will be written in the code
with the prefix np., which means we access the function inside the package
NumPy. The function Gramdet(A,B,C,D,E,F) computes the determinant of the Gram
matrix associated to $A,B,C,D,E,F\in\left[0,\pi\right]$.

⬇

1import numpy as np

2

3def Gramdet(A,B,C,D,E,F):

4G = matrix([ [1,-cos(A),-cos(B),-cos(F)],

5[-cos(A),1,-cos(C),-cos(E)],

6[-cos(B),-cos(C),1,-cos(D)],

7[-cos(F),-cos(E),-cos(D),1]])

8res = G.determinant()

9return res

Listing 1: The determinant of the Gram matrix of $T$

The U(z,A,B,C,D,E,F) function computes $U(z,T)$ for a given complex number $z$
and the dihedral angles $A,B,C,D,E$ and $F$ of a given generalized tetrahedron
$T$. We use the dilog(z) function for $z$ a complex number to compute
$\mathrm{Li_{2}}(z)$.

⬇

1def U(z,A,B,C,D,E,F):

2a = exp(I*A)

3b = exp(I*B)

4c = exp(I*C)

5d = exp(I*D)

6e = exp(I*E)

7f = exp(I*F)

8

9z1 = a*b*d*e*z

10z2 = a*c*d*f*z

11z3 = b*c*e*f*z

12z4 = -a*b*c*z

13z5 = -a*e*f*z

14z6 = -b*d*f*z

15z7 = -c*d*e*z

16

17res =
1/2*(dilog(z)+dilog(z1)+dilog(z2)+dilog(z3)-dilog(z4)-dilog(z5)-dilog(z6)-dilog(z7))

18return res

Listing 2: The complex valued fonction $U(z,T)$

Finally, the TetVolum(A,B,C,D,E,F) function uses the formula of Proposition
3.1 to compute the hyperbolic volume of a tetrahedron given its dihedral
angles $A$, $B$, $C$, $D$, $E$ and $F$.

⬇

1def TetVolum(A,B,C,D,E,F):

2a = exp(I*A)

3b = exp(I*B)

4c = exp(I*C)

5d = exp(I*D)

6e = exp(I*E)

7f = exp(I*F)

8

9det = Gramdet(A,B,C,D,E,F)

10

11zminus =
-2*((sin(A)*sin(D)+sin(B)*sin(E)+sin(C)*sin(F)-sqrt(det))/(a*d+b*e+c*f+a*b*f+a*c*e+b*c*d+d*e*f+a*b*c*d*e*f))

12

13zplus =
-2*((sin(A)*sin(D)+sin(B)*sin(E)+sin(C)*sin(F)+sqrt(det))/(a*d+b*e+c*f+a*b*f+a*c*e+b*c*d+d*e*f+a*b*c*d*e*f))

14

15res = imag((U(zminus,A,B,C,D,E,F)-U(zplus,A,B,C,D,E,F))/2)

16return res

Listing 3: The hyperbolic volume of $T$

####  5.1.2. Applying the volume formula on tetrahedra of $\mathcal{T}_{g}$

Following Frigerio’s computation of the complete angle structure detailed in
Section 4.3, we define the function HyperbolicVolume(g) to compute the
hyperbolic volume of $M_{g}$ given $g\in\\{2,3,4,...\\}$. Thanks to the
symmetries in the complete angle structure, this function computes the
hyperbolic volume of one tetrahedron and multiplies it by $2g+2$ (the number
of tetrahedra in the triangulation). We display several values for this
function for some values of $g$ in Figure 9.

⬇

1def HyperbolicVolume(g):

2Alphag=np.pi/(2*g+2)

3Betag=2*Alphag

4Gammag=arccos(1/(2*cos(Alphag)))

5Deltag=np.pi - 2*Gammag

6

7res=(2*g+2)*TetVolum(Gammag,Deltag,Gammag,Alphag,Betag,Alphag)

8return res

Listing 4: The hyperbolic volume of $M_{g}$ for a given $g$ $g$ | $\mathrm{Vol}(T_{g})$ | $\mathrm{Vol}(M_{g})$
---|---|---
2 | 2.007682006682397 | 12.046092040094381
3 | 2.2547631818606026 | 18.03810545488482
4 | 2.3603494908554774 | 23.603494908554772
5 | 2.415787949187158 | 28.989455390245897
6 | 2.448617485457304 | 34.28064479640226
7 | 2.469695490891516 | 39.51512785426426
8 | 2.484045062029212 | 44.71281111652581
9 | 2.494259571737797 | 49.88519143475594
10 | 2.5017908556003303 | 55.039398823207264
100 | 2.5369350366401 | 512.4608774013002
1000 | 2.5373497508910896 | 5079.774201283962
Figure 9. The hyperbolic volumes of $M_{g}$ and one tetrahedron $T_{g}$ of
$\mathcal{T}_{g}$ for various values of $g$.

###  5.2. Computing the Turaev–Viro invariants
$TV_{r,s}(M_{g},\mathcal{T}_{g})$

In this section, we will construct several functions in order to compute the
Turaev–Viro invariants $TV_{r,s}(M_{2},\mathcal{T}_{2})$ and
$TV_{r,s}(M_{3},\mathcal{T}_{3})$ given $(r,s)\in\mathbb{N}^{2}$ such that
$r\geq 3$ and $s\geq 1$. These functions can easily be extended to compute the
other Turaev–Viro invariants $TV_{r,s}(M_{g},\mathcal{T}_{g})$ for $g\geq 4$
(which we will do for $4\leqslant g\leqslant 7$).

We first import the NumPy package as in Section 5.1.1. We also import the
cmath package which gives us mathematical functions for complex numbers, such
as square roots for negative numbers. Functions imported from cmath will be
written with the prefix cm..

⬇

1import numpy as np

2import cmath as cm

Listing 5: Importing NumPy and cmath

The function quantum_number(r,s,n) returns the quantum number $[n]$ given $r$,
$s$ and $n\in\mathbb{N}$.

⬇

1def quantum_number(r,s,n):

2res=sin(s*n*(np.pi)/r)/sin(s*(np.pi)/r)

3return res

Listing 6: Quantum number

The function quantum_factorial(r,s,n) returns the factorial of a quantum
number $[n]!$ given $r$, $s$ and $n\in\mathbb{N}$.

⬇

1def quantum_factorial(r,s,n):

2if n == 0:

3return 1

4return prod([quantum_number(r,s,i) for i in range(1, n+1)])

Listing 7: Quantum factorial

The function q_big_delta_coeff(i,j,k,r,s) returns the coefficients
$\Delta(i,j,k)$ given $r$, $s$ and $(i,j,k)$ an admissible triple of elements
of $I_{r}$. Admissibility conditions ensure that $i+j-k$, $i+k-j$, $j+k-i$ and
$i+j+k+1$ are in $\mathbb{N}$ but since the values of $i$, $j$ and $k$ can be
half integers, the values $i+k-j$, $j+k-i$ and $i+j+k+1$ are stored as
rational type variable. To ensure that the function quantum_factorial() works,
we have to change their type from rational to integers using the $int()$
function. Since quantum_number() and thus quantum_factorial() can return
negative values, the term we take the square root of might thus be negative.
Hence we need to use the function cm.sqrt(), i.e the complex square root from
the cmath package.

⬇

1def q_big_delta_coeff(i,j,k,r,s):

2argsqrt = (quantum_factorial(r,s,int(i + j - k)) *
quantum_factorial(r,s,int(i + k - j)) * quantum_factorial(r,s,int(j + k - i)))
/ quantum_factorial(r,s,int(i + j + k + 1))

3

4res = cm.sqrt(argsqrt)

5return res

Listing 8: $\Delta$ coefficients

The function q_symbol(i, j, k, l, m, n, r, s) returns the quantum $6j$-symbol
$\begin{vmatrix}i&j&k\\\ l&m&n\end{vmatrix}$ given $r$, $s$ and
$(i,j,k,l,m,n)$ a sextuple of elements of $I_{r}$ such that $(i,j,k)$,
$(j,l,n)$, $(i,m,n)$ and $(k,l,m)$ are admissible.

⬇

1def q_symbol(i, j, k, l, m, n, r, s):

2prefac = q_big_delta_coeff(i, j, k, r, s) * q_big_delta_coeff(j, l, n, r, s)
* q_big_delta_coeff(i, m, n, r, s) * q_big_delta_coeff(k, l, m, r, s)

3

4zmin = max(i + j + k, j + l + n, i + m + n, k + l + m)

5zmax = min(i + j + l + m, i + k + l + n, j + k + m + n)

6

7sumres = 0

8for z in range(int(zmin), int(zmax) + 1):

9den = quantum_factorial(r,s,int(z - (i + j + k))) *
quantum_factorial(r,s,int(z - (j + l + n))) * quantum_factorial(r,s,int(z - (i
+ m + n))) * quantum_factorial(r,s,int(z - (k + l + m))) *
quantum_factorial(r,s,int(i + j + l + m - z)) * quantum_factorial(r,s,int(i +
k + l + n - z)) * quantum_factorial(r,s,int(j + k + m + n - z))

10

11sumres = sumres + (((-1) ** z) * quantum_factorial(r,s,int(z + 1))) / den

12

13res = prefac * sumres * sqrt(-1)**(int(2*(i+j+k+l+m+n)))

14return res

Listing 9: Quantum $6j$-symbols

The function edge(a,r,s) yields the term
$w_{a}:=(\scalebox{1.8}[1.0]{-}1)^{2a}[2a+1]$ for $r$, $s$ and $a\in I_{r}$.

⬇

1def edge(a,r,s) :

2res = ((-1)**(2*a))*quantum_number(r,s,int(2*a+1))

3return res

Listing 10: Edge term

The function term(a,b,c,g,r,s) computes one term of the sum in the Definition
3.7 applied to $(M_{g},\mathcal{T}_{g})$ given $r$, $s$ as before,
$g\in\\{2,3,4,...\\}$, $a\in I_{r}$, $b\in I_{r}$ and
$c=(c_{0},c_{1},c_{2},...,c_{g})\in I_{r}^{g+1}$ satisfying the conditions in
Theorem 4.8. This one term is equal to the product of
$w_{a}w_{b}w_{c_{0}}\ldots w_{c_{g}}$ with

$\scriptsize\begin{vmatrix}a&b&b\\\
c_{0}&c_{0}&c_{g}\end{vmatrix}\begin{vmatrix}a&b&b\\\
c_{0}&c_{0}&c_{1}\end{vmatrix}\ldots\begin{vmatrix}a&b&b\\\
c_{i}&c_{i}&c_{i-1}\end{vmatrix}\begin{vmatrix}a&b&b\\\
c_{i}&c_{i}&c_{i+1}\end{vmatrix}\ldots\begin{vmatrix}a&b&b\\\
c_{g}&c_{g}&c_{g-1}\end{vmatrix}\begin{vmatrix}a&b&b\\\
c_{g}&c_{g}&c_{0}\end{vmatrix}.$

⬇

1def term(a,b,c,g,r,s):

2res=edge(a, r, s)*edge(b, r, s)

3

4for i in range(g+1):

5if i==0:

6res = res * edge(c[i], r, s) * q_symbol(a, b, b, c[i], c[i], c[g], r, s) *
q_symbol(a, b, b, c[i], c[i], c[i+1], r, s)

7elif i==g:

8res = res * edge(c[i], r, s) * q_symbol(a, b, b, c[i], c[i], c[i-1], r, s) *
q_symbol(a, b, b, c[i], c[i], c[0], r, s)

9else:

10res = res * edge(c[i], r, s) * q_symbol(a, b, b, c[i], c[i], c[i-1], r, s) *
q_symbol(a, b, b, c[i], c[i], c[i+1], r, s)

11

12return res

Listing 11: One term of the sum in $TV_{r,s}(M_{g},\mathcal{T}_{g})$ for a
given admissible coloring

The function turaevvirog2(r,s) computes $TV_{r,s}(M_{2},\mathcal{T}_{2})$,
while the function turaevvirog3(r,s) computes
$TV_{r,s}(M_{3},\mathcal{T}_{3})$ given $r$ and $s$. These two functions
consists in listing all the admissible colorings in the sense of Theorem 4.8
and computing the associated term contributing to the sum in
$TV_{r,s}(M_{g},\mathcal{T}_{g})$ via the term() function for $g\in\\{2,3\\}$.

We now detail the reasoning behind the following code. Let
$a,b,c_{0},...,c_{g}\in I_{r}$.

The $a$ loop: For the coloring to be admissible, $a$ has to satisfy:

  * (1)

$a\in\mathbb{N}$.

The first loop is over the label $a$ which covers all integers from $0$ to
$\frac{r-2}{2}$. We will refer to it as the $a$ loop. The floor() function
ensures that the loop ends at $\left\lfloor{\frac{r-2}{2}}\right\rfloor$.

The $b$ loop: For $a$ fixed and an admissible coloring, $b$ has to satisfy:

  * (2)

$b\in\mathbb{N}$,

  * (4)

$\frac{a}{2}\leq b\leq\frac{r-2-a}{2}$.

We thus create, inside the $a$ loop, a second loop over the label $b$ (called
the $b$ loop), which covers all integers from $\frac{a}{2}$ to
$\frac{r-2-a}{2}$. The floor() and ceil() functions ensure that the loop ends
at $\left\lfloor{\frac{r-2-a}{2}}\right\rfloor$ and starts at
$\left\lceil{\frac{a}{2}}\right\rceil$.

The two kinds of nested loops inside the $b$ loop: For an admissible coloring,
the labels $c_{i}$ (for $i\in\\{0,1,2,...,g\\}$) have to satisfy:

  * (3)

either ($\forall i\in\\{0,1,2,...,g\\}$, $c_{i}\in\mathbb{N}$) or ($\forall
i\in\\{0,1,2,...,g\\}$, $c_{i}\in\frac{\mathbb{N}_{odd}}{2}$).

We denote those two categories as integer states and half-integer states.
Thus, we create two different loops inside the $b$ loop (one for each).

The first family of nested loops, for integer states: Let us assume ($\forall
i\in\\{0,\ldots,g\\}$, $c_{i}\in\mathbb{N}$). For a fixed $b$ and an
admissible coloring, the label $c_{0}$ has to satisfy:

  * (6a)

$\frac{b}{2}\leq c_{0}\leq\frac{r-2-b}{2}$.

We thus create, inside the $b$ loop, a loop over the label $c_{0}$, the
$c_{0}$ loop, which covers all integers from
$\left\lceil{\frac{b}{2}}\right\rceil$ to
$\left\lfloor{\frac{r-2-b}{2}}\right\rfloor$. This $c_{0}$ loop will contain a
new $c_{1}$ loop, which will in turn contain a $c_{2}$ loop, and so on until a
final $c_{g}$ loop. Let us now detail how this induction works. For
$i\in\\{0,...,g-1\\}$, for fixed $a$, $b$, $c_{i}$ and an admissible coloring,
the label $c_{i+1}$ has to satisfy the following conditions:

  * (5a)

$\frac{b}{2}\leq c_{i+1}\leq\frac{r-2-b}{2}$,

  * (5b)

$a-c_{i}\leq c_{i+1}\leq r-2-a-c_{i}$,

  * (5c-d)

$c_{i}-\min(a,b)\leq c_{i+1}\leq c_{i}+\min(a,b)$.

For all $i\in\\{0,...,g-1\\}$, we create in the loop $c_{i}$ two variables
$m_{i+1}$ and $M_{i+1}$: $m_{i+1}$ is the maximum of the three lower bounds on
$c_{i+1}$ and $M_{i+1}$ is the minimum of the three upper bounds on $c_{i+1}$.
We thus create, inside the $c_{i}$ loop, a loop over the label $c_{i+1}$
(called the $c_{i+1}$ loop), which ranges to all integers from
$\left\lceil{m_{i+1}}\right\rceil$ to $\left\lfloor{M_{i+1}}\right\rfloor$.
Finally, for fixed $a$, $b$, $c_{0},\ldots,c_{g}$, for the coloring to be
admissible, the following conditions also need to be satisfied:

  * (6b)

$a-c_{g}\leq c_{0}\leq r-2-a-c_{g}$,

  * (6c-d)

$c_{g}-\min(a,b)\leq c_{0}\leq c_{g}+\min(a,b)$.

Hence we add an if loop that checks those two conditions. If they are
satisfied, we then compute the term associated to the coloring
$(a,b,c_{0},c_{1},...,c_{g})$ via term().

The second family of nested loops, for half-integer states: Once we have
covered all the integer states, we create inside the $b$ loop, a second loop
over the label $c_{0}$, which covers all half-integers from
$\left\lfloor{\frac{b}{2}}\right\rfloor+\frac{1}{2}$ to
$\left\lceil{\frac{r-2-b}{2}}\right\rceil-\frac{1}{2}$. The function
np.arange() allows us to start and end loops at non-integer values and specify
the step of the loop to be 1. The rest of this new $c_{0}$ loop is constructed
in the same way as for the $c_{0}$ loop for integer states, but the nested
loops are over half-integers instead of integers.

Conclusion: As proved in Theorem 4.8, these loops go over all admissible
states, thus we obtain a numerical value of the exact formula
$TV_{r,s}(M_{g},\mathcal{T}_{g})$ for $g\in\\{2,3\\}$. In the same way, one
can construct a function which computes $TV_{r,s}(M_{g},\mathcal{T}_{g})$ for
$g\geq 4$ (one would simply need to add more nested loops in the two
families). In the current project, we did exactly this: we defined the
functions turaevvirog4(r,s), turaevvirog5(r,s), turaevvirog6(r,s) and
turaevvirog7(r,s), but for the sake of brevity we do not write their codes
here (their codes can be easily guessed from the detailed examples of
turaevvirog2(r,s) and turaevvirog3(r,s)).

⬇

1def turaevvirog2(r,s):

2res= 0

3for a in range(floor((r-2)/2)+1):

4for b in range(ceil(a/2),floor((r-2-a)/2)+1):

5m=min(a,b)

6for c0 in range(ceil(b/2),floor((r-2-b)/2)+1):

7m1=max(c0-m,a-c0,b/2)

8M1=min(c0+m,r-2-a-c0,(r-2-b)/2)

9for c1 in range(ceil(m1),floor(M1)+1):

10m2=max(c1-m,a-c1,b/2)

11M2=min(c1+m,r-2-a-c1,(r-2-b)/2)

12for c2 in range(ceil(m2),floor(M2)+1):

13if (-m<=c2-c0<=m) and (a<=c2+c0<=r-2-a):

14c=[c0,c1,c2]

15res=res+term(a,b,c,2,r,s)

16

17for c0 in np.arange(floor(b/2)+1/2,ceil((r-2-b)/2),1):

18m1=max(c0-m,a-c0,b/2)

19M1=min(c0+m,r-2-a-c0,(r-2-b)/2)

20for c1 in np.arange(floor(m1)+1/2,ceil(M1),1):

21m2=max(c1-m,a-c1,b/2)

22M2=min(c1+m,r-2-a-c1,(r-2-b)/2)

23for c2 in np.arange(floor(m2)+1/2,ceil(M2),1):

24if (-m<=c2-c0<=m) and (a<=c2+c0<=r-2-a):

25c=[c0,c1,c2]

26res=res+term(a,b,c,2,r,s)

27return res

Listing 12: $TV_{r,s}(M_{2},\mathcal{T}_{2})$

⬇

1def turaevvirog3(r,s):

2res= 0

3for a in range(floor((r-2)/2)+1):

4for b in np.arange(ceil(a/2),floor((r-2-a)/2)+1,1):

5m=min(a,b)

6for c0 in np.arange(ceil(b/2),floor((r-2-b)/2)+1,1):

7m1=max(c0-m,a-c0,b/2)

8M1=min(c0+m,r-2-a-c0,(r-2-b)/2)

9for c1 in np.arange(ceil(m1),floor(M1)+1,1):

10m2=max(c1-m,a-c1,b/2)

11M2=min(c1+m,r-2-a-c1,(r-2-b)/2)

12for c2 in np.arange(ceil(m2),floor(M2)+1,1):

13m3=max(c2-m,a-c2,b/2)

14M3=min(c2+m,r-2-a-c2,(r-2-b)/2)

15for c3 in np.arange(ceil(m3),floor(M3)+1,1):

16if (-m<=c3-c0<=m) and (a<=c3+c0<=r-2-a):

17c=[c0,c1,c2,c3]

18res=res+term(a,b,c,3,r,s)

19

20for c0 in np.arange(floor(b/2)+1/2,ceil((r-2-b)/2),1):

21m1=max(c0-m,a-c0,b/2)

22M1=min(c0+m,r-2-a-c0,(r-2-b)/2)

23for c1 in np.arange(floor(m1)+1/2,ceil(M1),1):

24m2=max(c1-m,a-c1,b/2)

25M2=min(c1+m,r-2-a-c1,(r-2-b)/2)

26for c2 in np.arange(floor(m2)+1/2,ceil(M2),1):

27m3=max(c2-m,a-c2,b/2)

28M3=min(c2+m,r-2-a-c2,(r-2-b)/2)

29for c3 in np.arange(floor(m3)+1/2,ceil(M3),1):

30if (-m<=c3-c0<=m) and (a<=c3+c0<=r-2-a):

31c=[c0,c1,c2,c3]

32res=res+term(a,b,c,3,r,s)

33return res

Listing 13: $TV_{r,s}(M_{3},\mathcal{T}_{3})$

###  5.3. Asymptotic behavior of $QV_{r,2}(M_{2})$ and $QV_{r,2}(M_{3})$

The function quantumvirog2(r,s) computes
$QV_{r,s}(M_{2})=\frac{s\pi}{r-2}\log\left(TV_{r,s}(M_{2},\mathcal{T}_{2})\right)$
given $(r,s)\in\mathbb{N}^{2}$ such that $r\geq 3$ and $s\geq 1$.

⬇

1def quantumvirog2(r,s):

2res=(s*(np.pi)/(r-2))*log(turaevvirog2(r,s))

3return res

Listing 14: $QV_{r,s}(M_{2},\mathcal{T}_{2})$

The function quantumvirog3(r,s) computes
$QV_{r,s}(M_{3})=\frac{s\pi}{r-2}\log\left(TV_{r,s}(M_{3},\mathcal{T}_{3})\right)$
$(r,s)\in\mathbb{N}^{2}$ such that $r\geq 3$ and $s\geq 1$.

⬇

1def quantumvirog3(r,s):

2res=(s*(np.pi)/(r-2))*log(turaevvirog3(r,s))

3return res

Listing 15: $QV_{r,s}(M_{3},\mathcal{T}_{3})$

Similarly, the definitions of quantumvirog4(r,s), quantumvirog5(r,s),
quantumvirog6(r,s) and quantumvirog7(r,s) follow immediately.

Let us recall that we aim for a numerical test of Conjecture 1.1. Thus, we set
$s=2$ in order to numerically compute $QV_{r,2}(M_{2})$ and $QV_{r,2}(M_{3})$.

Note that, for some $r$ it happens that $TV_{r,2}(M_{2},\mathcal{T}_{2})$ is
negative. Since we require the argument of the complex logarithm function to
be in $[0,2\pi[$, the imaginary part of $QV_{r,2}(M_{2})$ is either 0 when
$TV_{r,2}(M_{2},\mathcal{T}_{2})$ is positive or $\frac{2\pi^{2}}{r}$ when
$TV_{r,2}(M_{2},\mathcal{T}_{2})$ is negative, which converges to $0$ as
$r\to\infty$. Therefore to test the convergence of $QV_{r,2}(M_{2})$ we can
forget about imaginary parts and consider only the real parts.

We compute the function quantumvirog2(r,s) for $s=2$ and increasing values of
$r$, and we obtain the table of values of
$\mathcal{R}\left(QV_{r,2}(M_{2})\right)$ shown in Figure 10. We use the
numerical approximation .n() with the best available precision (around prec =
180).

⬇

1quantumvirog2(r,2).real().n(prec=180)

Listing 16: Numerical approximation of
$\mathcal{R}\left(QV_{r,2}(M_{2})\right)$

Similarly, we use the functions quantumvirog3(r,s), …, quantumvirog7(r,s) for
$s=2$ and increasing values of $r$, and we display the values of
$\mathcal{R}\left(QV_{r,2}(M_{g})\right)$ in Figure 10.

As we will detail in Section 6, we obtain surprising values for high $r$ and
$g=2,3$, probably due to numerical errors. Those values are written in red in
Figure 10.

| $r$ | $\mathcal{R}\left(QV_{r,2}(M_{2})\right)$
---|---
5 | 8.14385123663626
7 | 9.18650442759997
9 | 9.65004427173429
11 | 9.96879239401443
13 | 10.20513879726808
15 | 10.38914324592799
17 | 10.53704472005768
19 | 10.65879117905018
21 | 10.76091340012164
23 | 10.84790597624064
25 | 10.92297357052110
27 | 10.98846715752597
29 | 11.04614827534519
31 | 11.09819658700029
33 | 11.10744853337351
35 | 10.85076510281595
37 | 11.70823932238226
39 | 12.05034471052339
41 | 12.57984278565481
43 | 13.01497045469742
45 | 13.57883304172589
47 | 13.99851452347661
| $r$ | $\mathcal{R}\left(QV_{r,2}(M_{3})\right)$
---|---
5 | 11.49177317419101
7 | 12.80934693191113
9 | 13.58615197340893
11 | 14.12955507845825
13 | 14.53997951590672
15 | 14.86388896169300
17 | 15.12724763049115
19 | 15.34618602238218
21 | 15.53141775410042
23 | 15.69039789582600
25 | 15.82849506550996
27 | 15.94972272572273
29 | 16.05847664488577
31 | 16.12064941438458
33 | 16.64108419344305
35 | 17.23677472848113
37 | 17.65793100469928
39 | 18.19438875927008
| $r$ | $\mathcal{R}\left(QV_{r,2}(M_{4})\right)$
---|---
5 | 14.51784517894469
7 | 16.30280237431099
9 | 17.32714285662395
11 | 18.05414567452926
13 | 18.60945703261760
15 | 19.05151621992931
17 | 19.41350816169271
19 | 19.71628402919349
21 | 19.97380655712918
23 | 20.19586182173212
25 | 20.38962564202214
27 | 20.54717170623221
| $r$ | $\mathcal{R}\left(QV_{r,2}(M_{5})\right)$
---|---
5 | 17.56864290428003
7 | 19.74442367439225
9 | 20.99442151342528
11 | 21.88836919170208
13 | 22.57622952582667
15 | 23.12700521166837
17 | 23.58015181610567
19 | 23.96067740594393
21 | 24.28544874705841
23 | 24.56622464869820
| $r$ | $\mathcal{R}\left(QV_{r,2}(M_{6})\right)$
---|---
5 | 20.59635740610918
7 | 23.16334886690935
9 | 24.62826235095652
11 | 25.68044858255137
13 | 26.49408736663125
15 | 27.14829604792329
17 | 27.68837084809290
19 | 28.14316996246829
21 | 28.53221301857429
23 | 28.85466729936771
| $r$ | $\mathcal{R}\left(QV_{r,2}(M_{7})\right)$
---|---
5 | 23.62294303366446
7 | 26.57176683519978
9 | 28.24541308192440
11 | 29.45065948405797
13 | 30.38589828885670
15 | 31.14019388548824
17 | 31.76448809338449
19 | 32.29128792277911
Figure 10. Values of $\mathcal{R}\left(QV_{r,2}(M_{g})\right)$ for $2\leqslant
g\leqslant 7$ and $r\geqslant 5$ (assumed numerical errors are written in
red).

In the following code, we look for the best interpolation of our data for the
values $\mathcal{R}\left(QV_{r,2}(M_{2})\right)$, when $5\leqslant r\leqslant
33$. We look for a model of the form $a+b\frac{\ln(r-2)}{r-2}+c\frac{1}{r-2}$
(as in Conjecture 1.2). We use the function find_fit.

⬇

1data=[(2*i+1,quantumvirog2(2*i+1,2).real().n(prec=180)) for i in range(2,15)]

2

3var(’a, b, c, r’)

4model(r)= a+ b*2*pi*ln(r-2)/(r-2) + c/(r-2)

5

6sol = find_fit(data,model)

7show(sol)

Listing 17: 3-term interpolation of $\mathcal{R}\left(QV_{r,2}(M_{2})\right)$
for $5\leqslant r\leqslant 33$.

We find the values

$a=11.86209740389381,b=-0.835561949347834,c=-5.310168450722084,$

and in particular a constant term $a$ equal to the expected hyperbolic volume
up to

$\dfrac{12.046092040094381-11.86209740389381}{12.046092040094381}\approx
1.5\%.$

We then do the same for $M_{3}$, in the following code.

⬇

1data=[(2*i+1,quantumvirog3(2*i+1,2).real().n(prec=180)) for i in range(2,14)]

2

3

4var(’a, b, c, r’)

5model(r)= a+ b*2*pi*ln(r-2)/(r-2) + c/(r-2)

6

7sol = find_fit(data,model)

8show(sol)

Listing 18: 3-term interpolation of $\mathcal{R}\left(QV_{r,2}(M_{3})\right)$
for $5\leqslant r\leqslant 31$.

We find the values

$a=17.712568980467715,b=-1.95506206171866,c=-5.092760978446523,$

and in particular a constant term $a$ equal to the expected hyperbolic volume
up to

$\dfrac{18.03810545488482-17.712568980467715}{18.03810545488482}\approx
1.8\%.$

For each $g\in\\{4,\ldots 7\\}$, we interpolate all available values of
$\mathcal{R}\left(QV_{r,2}(M_{g})\right)$ with the same model (since no
numerical strangeness occur in these cases). The values for $a,b,c$ are listed
in Figure 11.

$g$ | $r_{max}$ | $\mathrm{Vol}(M_{g})$ | $a$ | $b$ | $c$ | $\frac{\mathrm{Vol}(M_{g})-a}{\mathrm{Vol}(M_{g})}$
---|---|---|---|---|---|---
2 | 33 | 12.04609204 | 11.86209740 | -0.83556194 | -5.31016845 | $\approx 1.5\%$
3 | 31 | 18.03810545 | 17.71256898 | -1.95506206 | -5.09276097 | $\approx 1.8\%$
4 | 27 | 23.60349490 | 22.91592390 | -2.65679563 | -6.74587906 | $\approx 2.9\%$
5 | 23 | 28.98945539 | 27.83557719 | -3.23491649 | -8.35921398 | $\approx 3.9\%$
6 | 23 | 34.28064479 | 32.73892860 | -3.85245863 | -9.69525194 | $\approx 4.5\%$
7 | 19 | 39.51512785 | 37.25645299 | -4.15342419 | -12.1205935 | $\approx 5.7\%$
Figure 11. Values of the interpolating coefficients $a,b,c$ for the model
$a+b\cdot\frac{2\pi\ln(r-2)}{r-2}+c\frac{1}{r-2}$ for
$\mathcal{R}\left(QV_{r,2}(M_{g})\right)$, with $5\leqslant r\leqslant
r_{max}$.

##  6\. Numerical Results

###  6.1. The case of $M_{2}$

Let us now state the relevant structures and invariants of $M_{2}$, which will
then allow us to numerically check the volume conjecture for this manifold.

####  6.1.1. Triangulation

Figure 12 displays the ideal triangulation $\mathcal{T}_{g}$ of $M_{g}$ in the
case $g=2$ (note that some gluing information is not directly stated in the
picture for clarity). The $0$-skeleton $(\mathcal{T}_{2})^{0,\sim}$ has two
elements $\nu_{1}$ (corresponding to the toroidal boundary component) and
$\nu_{2}$ (corresponding to the boundary component of genus $2$). The
$1$-skeleton $(\mathcal{T}_{2})^{1,\sim}$ contains five classes
$\eta_{1},\ldots,\eta_{5}$.

$\nu_{2}$$\nu_{2}$$\nu_{1}$$\nu_{2}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta_{1}$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\eta_{5}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\eta_{3}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\eta_{3}$

$\nu_{2}$$\nu_{2}$$\nu_{2}$$\nu_{1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta_{1}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\eta_{3}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\eta_{4}$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\eta_{3}$

$\nu_{2}$$\nu_{2}$$\nu_{1}$$\nu_{2}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta_{1}$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\eta_{3}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\eta_{4}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\eta_{4}$

$\nu_{2}$$\nu_{2}$$\nu_{2}$$\nu_{1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta_{1}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\eta_{4}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\eta_{5}$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\eta_{4}$

$\nu_{2}$$\nu_{2}$$\nu_{1}$$\nu_{2}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta_{1}$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\eta_{4}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\eta_{5}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\eta_{5}$

$\nu_{2}$$\nu_{2}$$\nu_{2}$$\nu_{1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta_{1}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\eta_{5}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\eta_{3}$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\eta_{5}$

Figure 12. The ordered ideal triangulation $\mathcal{T}_{2}$ of the 3-manifold
$M_{2}$

####  6.1.2. Hyperbolic structure

As a specific case of Section 4.3, the unique complete hyperbolic structure on
the manifold $M_{2}$ is given by the angles

$\alpha_{2}=\frac{\pi}{6},\ \ \beta_{2}=2\alpha_{2}=\frac{\pi}{3},\ \
\gamma_{2}=\arccos((2\cos\alpha_{2})^{-1}),\ \ \delta_{2}=\pi-2\gamma_{2},$

and the hyperbolic volume of $M_{2}$ is computed (via the code of Section
5.1.2) to be $\mathrm{Vol}(M_{2})=12.046092040094381...$ (which corresponds to
the value computed with Orb in [8]).

####  6.1.3. Admissible colorings and Turaev–Viro Invariants

From Definition 3.7, we compute the edge terms and tetrahedron terms
contributing to $TV_{r,s}(M_{2},\mathcal{T}_{2})$. Since $\mathcal{T}_{2}$ has
no regular vertices, the regular vertices term is thus $N=\left(\sum_{i\in
I_{r}}w_{i}^{2}\right)^{0}=1$.

0231$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}c_{2}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}c_{0}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}c_{0}$$\begin{vmatrix}b&a&b\\\
c_{0}&c_{0}&c_{2}\end{vmatrix}$

0213$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}c_{0}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}c_{1}$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}c_{0}$$\begin{vmatrix}b&a&b\\\
c_{1}&c_{0}&c_{0}\end{vmatrix}$

0231$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}c_{0}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}c_{1}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}c_{1}$$\begin{vmatrix}b&a&b\\\
c_{1}&c_{1}&c_{0}\end{vmatrix}$

0213$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}c_{1}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}c_{2}$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}c_{1}$$\begin{vmatrix}b&a&b\\\
c_{2}&c_{1}&c_{1}\end{vmatrix}$

0231$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}c_{1}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}c_{2}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}c_{2}$$\begin{vmatrix}b&a&b\\\
c_{2}&c_{2}&c_{1}\end{vmatrix}$

0213$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}c_{2}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}c_{0}$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}c_{2}$$\begin{vmatrix}b&a&b\\\
c_{0}&c_{2}&c_{2}\end{vmatrix}$

Figure 13. The coloring $c$ of the ideal tetrahedra of $\mathcal{T}_{2}$
together with their respective tetrahedron terms

Let $(a,b,c_{0},c_{1},c_{2})$ be a quintuple of elements of $I_{r}$. A
coloring $c:X^{1}_{\sim}\to I_{r}$ such as in Figure 13 is admissible if and
only if it satisfies the conditions of Theorem 4.8, thus

$\mathcal{A}_{r}(M_{2},\mathcal{T}_{2})=\left\\{\begin{pmatrix}a\\\ b\\\
c_{0}\\\ c_{1}\\\ c_{2}\end{pmatrix}\in
I_{r}^{5}\;\begin{tabular}[]{|l}$a,b\in\mathbb{N}$,\\\ $\frac{a}{2}\leq
b\leq\frac{r-2-a}{2}$,\\\ either $c_{0},c_{1},c_{2}\in\mathbb{N}$ or
$c_{0},c_{1},c_{2}\in\frac{\mathbb{N}_{odd}}{2}$,\\\
$\max\left(\frac{b}{2},a-c_{3},c_{3}-\min(a,b)\right)\leq c_{0}$,\\\
$c_{0}\leq\min\left(\frac{r-2-b}{2},r-2-a-c_{3},\min(a,b)+c_{3}\right)$,\\\
$\max\left(\frac{b}{2},a-c_{0},c_{0}-\min(a,b)\right)\leq c_{1}$,\\\
$c_{1}\leq\min\left(\frac{r-2-b}{2},r-2-a-c_{0},\min(a,b)+c_{0}\right)$,\\\
$\max\left(\frac{b}{2},a-c_{1},c_{1}-\min(a,b)\right)\leq c_{2}$,\\\
$c_{2}\leq\min\left(\frac{r-2-b}{2},r-2-a-c_{1},\min(a,b)+c_{1}\right)$.\end{tabular}\right\\}.$

From Figure 13, we can determine the six tetrahedron terms $|T|_{c}$ and five
edge terms $|\eta|_{c}$ associated to the coloring $c$ which gives us the
following equation:

$\displaystyle
TV_{r,s}(M_{2},\mathcal{T}_{2})=\sum_{(a,b,c_{0},c_{1},c_{2})\in\mathcal{A}_{r}(M_{2},\mathcal{T}_{2})}w_{a}w_{b}w_{c_{0}}w_{c_{1}}w_{c_{2}}\cdot$
$\displaystyle\begin{vmatrix}b&a&b\\\
c_{0}&c_{0}&c_{2}\end{vmatrix}\begin{vmatrix}b&a&b\\\
c_{1}&c_{0}&c_{0}\end{vmatrix}$ $\displaystyle\cdot$
$\displaystyle\begin{vmatrix}b&a&b\\\
c_{1}&c_{1}&c_{0}\end{vmatrix}\begin{vmatrix}b&a&b\\\
c_{2}&c_{1}&c_{1}\end{vmatrix}$ $\displaystyle\cdot$
$\displaystyle\begin{vmatrix}b&a&b\\\
c_{2}&c_{2}&c_{1}\end{vmatrix}\begin{vmatrix}b&a&b\\\
c_{0}&c_{2}&c_{2}\end{vmatrix}.$

Using Proposition 3.6 and a bit of reordering, we can rewrite the summation
as:

$\displaystyle
TV_{r,s}(M_{2},\mathcal{T}_{2})=\sum_{(a,b,c_{0},c_{1},c_{2})\in\mathcal{A}_{r}(M_{2},\mathcal{T}_{2})}w_{a}w_{b}w_{c_{0}}w_{c_{1}}w_{c_{2}}\cdot$
$\displaystyle\begin{vmatrix}a&b&b\\\
c_{0}&c_{0}&c_{2}\end{vmatrix}\begin{vmatrix}a&b&b\\\
c_{0}&c_{0}&c_{1}\end{vmatrix}$ $\displaystyle\cdot$
$\displaystyle\begin{vmatrix}a&b&b\\\
c_{1}&c_{1}&c_{0}\end{vmatrix}\begin{vmatrix}a&b&b\\\
c_{1}&c_{1}&c_{2}\end{vmatrix}$ $\displaystyle\cdot$
$\displaystyle\begin{vmatrix}a&b&b\\\
c_{2}&c_{2}&c_{1}\end{vmatrix}\begin{vmatrix}a&b&b\\\
c_{2}&c_{2}&c_{0}\end{vmatrix}.$

####  6.1.4. Numerical check of the volume conjecture

$0$$10$$20$$30$$40$$50$$8$$10$$12$$14$$r$$\mathcal{R}\left(QV_{r,2}(M_{2})\right)$Vol$(M_{2})=12.046092...$$11.86-5.25\frac{\ln(r-2)}{r-2}-5.31\frac{1}{r-2}$
Figure 14. Graph of the values of $\mathcal{R}\left(QV_{r,2}(M_{2})\right)$
(for $5\leqslant r\leqslant 47$, blue dots) compared with the hyperbolic
volume $\mathrm{Vol}(M_{2})$ (dashed and blue) and an interpolation of
$\mathcal{R}\left(QV_{r,2}(M_{2})\right)$ for $r\leqslant 33$ (blue curve).

Recall that we compute and study the behavior of
$QV_{r,2}(M):=\frac{2\pi}{r-2}\log\left(TV_{r,2}(M)\right)$. Figure 14
displays the values of $\mathcal{R}\left(QV_{r,2}(M_{2})\right)$ for
$5\leqslant r\leqslant 47$ (blue dots), computed with maximal available
precision via the code of Section 5.2.

For $r\leqslant 33$, we observe the expected convergence to the hyperbolic
volume $\mathrm{Vol}(M_{2})$ (displayed with the blue dashed line); more
precisely, when we fit the data for $5\leqslant r\leqslant 33$ with the model
$a+b\frac{\ln(r-2)}{r-2}+c\frac{1}{r-2}$ (see Section 5.3), we find a constant
term which is very close to $\mathrm{Vol}(M_{2})$ (up to $1.5\%$), and the
interpolating function found by SageMath (displayed in a full blue curve)
appears to be close to the blue dots.

However, a strange behavior starts at $r=35$, and the values of
$\mathcal{R}\left(QV_{r,2}(M_{2})\right)$ break the expected pattern. We offer
a possible explanation that is still compatible with Conjecture 3.8: since the
Turaev–Viro invariants are computed as sums of a large set of terms
(exponentially many in $r$) which may have vastly different orders of
magnitude, the numerical approximations of the computer might truncate away
subtle parts of the terms in the sum, which translates to larger and larger
errors in the final results.

To test this theory, we computed the same values with smaller precision, on
the same machine, expecting to find different values for high $r$. However, it
was not the case. Nevertheless, we observed that a different computer yielded
slightly different numerical values than the one stated in this paper, the
difference increasing as $r$ grew larger.

Such numerical errors for large $r$ seem unavoidable when computing
Turaev–Viro invariants, especially for triangulations with a large number of
tetrahedra. One can for instance compare the maximum values of $r$ that were
studied for triangulations with less than $4$ tetrahedra in [3], and the ones
in this paper, where $M_{2}$ has $6$ tetrahedra, $M_{3}$ has $8$ tetrahedra,
and so on.

If we restrict to the values for $r\leqslant 33$, Conjectures 1.1 and 1.2 seem
satisfied for $M_{2}$, as we observe the expected asymptotic behavior with
small deviations.

###  6.2. The case of $M_{3}$

####  6.2.1. Triangulation

Figure 15 displays the ideal triangulation $\mathcal{T}_{g}$ of $M_{g}$ in the
case $g=3$. The $0$-skeleton $(\mathcal{T}_{3})^{0,\sim}$ has two elements
$\nu_{1}$ (corresponding to the toroidal boundary component) and $\nu_{2}$
(corresponding to the boundary component of genus $3$). The $1$-skeleton
$(\mathcal{T}_{3})^{1,\sim}$ contains six classes $\eta_{1},\ldots,\eta_{6}$.

$\nu_{2}$$\nu_{2}$$\nu_{1}$$\nu_{2}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta_{1}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta_{6}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\eta_{3}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\eta_{3}$

$\nu_{2}$$\nu_{2}$$\nu_{2}$$\nu_{1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta_{1}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\eta_{3}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\eta_{4}$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\eta_{3}$

$\nu_{2}$$\nu_{2}$$\nu_{1}$$\nu_{2}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta_{1}$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\eta_{3}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\eta_{4}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\eta_{4}$

$\nu_{2}$$\nu_{2}$$\nu_{2}$$\nu_{1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta_{1}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\eta_{4}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\eta_{5}$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\eta_{4}$

$\nu_{2}$$\nu_{2}$$\nu_{1}$$\nu_{2}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta_{1}$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\eta_{4}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\eta_{5}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\eta_{5}$

$\nu_{2}$$\nu_{2}$$\nu_{2}$$\nu_{1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta_{1}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\eta_{5}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta_{6}$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\eta_{5}$

$\nu_{2}$$\nu_{2}$$\nu_{1}$$\nu_{2}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta_{1}$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\eta_{5}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta_{6}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta_{6}$

$\nu_{2}$$\nu_{2}$$\nu_{2}$$\nu_{1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta_{1}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta_{6}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\eta_{2}$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\eta_{3}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta_{6}$

Figure 15. The ordered ideal triangulation $\mathcal{T}_{3}$ of the 3-manifold
$M_{3}$

####  6.2.2. Hyperbolic structure

As a specific case of Section 4.3, the unique complete hyperbolic structure on
the manifold $M_{3}$ is given by the angles

$\alpha_{3}=\frac{\pi}{8},\ \ \beta_{3}=2\alpha_{3}=\frac{\pi}{4},\ \
\gamma_{3}=\arccos((2\cos\alpha_{3})^{-1}),\ \ \delta_{3}=\pi-2\gamma_{3},$

and the hyperbolic volume of $M_{3}$ is computed (via the code of Section
5.1.2) to be $\mathrm{Vol}(M_{3})=18.03810545488482...$.

####  6.2.3. Admissible colorings and Turaev–Viro Invariants

Using Definition 3.7 for $(M_{3},\mathcal{T}_{3})$, we can determine the edge
terms and tetrahedron terms contributing to $TV_{r,s}(M_{3},\mathcal{T}_{3})$.
Since there are no regular vertices in $\mathcal{T}_{3}$, the regular vertices
term is thus $N=\left(\sum_{i\in I_{r}}w_{i}^{2}\right)^{0}=1$.

0231$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{3}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}c_{0}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}c_{0}$$\begin{vmatrix}b&a&b\\\
c_{0}&c_{0}&c_{3}\end{vmatrix}$

0213$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}c_{0}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}c_{1}$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}c_{0}$$\begin{vmatrix}b&a&b\\\
c_{1}&c_{0}&c_{0}\end{vmatrix}$

0231$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}c_{0}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}c_{1}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}c_{1}$$\begin{vmatrix}b&a&b\\\
c_{1}&c_{1}&c_{0}\end{vmatrix}$

0213$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}c_{1}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}c_{2}$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}c_{1}$$\begin{vmatrix}b&a&b\\\
c_{2}&c_{1}&c_{1}\end{vmatrix}$

0231$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a$$\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}c_{1}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}c_{2}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}c_{2}$$\begin{vmatrix}b&a&b\\\
c_{2}&c_{2}&c_{1}\end{vmatrix}$

0213$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}c_{2}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{3}$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}c_{2}$$\begin{vmatrix}b&a&b\\\
c_{3}&c_{2}&c_{2}\end{vmatrix}$

0231$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a$$\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}c_{2}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{3}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{3}$$\begin{vmatrix}b&a&b\\\
c_{3}&c_{3}&c_{2}\end{vmatrix}$

0213$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{3}$$\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}b$$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}c_{0}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{3}$$\begin{vmatrix}b&a&b\\\
c_{0}&c_{3}&c_{3}\end{vmatrix}$

Figure 16. The coloring $c$ of the ideal tetrahedra of $\mathcal{T}_{3}$
together with their respective tetrahedron terms

Let $(a,b,c_{0},c_{1},c_{2},c_{3})$ be a sextuple of elements of $I_{r}$. A
coloring $c:X^{1}_{\sim}\to I_{r}$ such as in Figure 16 is admissible if and
only if it satisfies the conditions of Theorem 4.8, thus

$\mathcal{A}_{r}(M_{3},\mathcal{T}_{3})=\left\\{\begin{pmatrix}a\\\ b\\\
c_{0}\\\ c_{1}\\\ c_{2}\\\ c_{3}\end{pmatrix}\in
I_{r}^{5}\;\begin{tabular}[]{|l}$a,b\in\mathbb{N}$,\\\ $\frac{a}{2}\leq
b\leq\frac{r-2-a}{2}$,\\\ either $c_{0},c_{1},c_{2}\in\mathbb{N}$ or
$c_{0},c_{1},c_{2}\in\frac{\mathbb{N}_{odd}}{2}$,\\\
$\max\left(\frac{b}{2},a-c_{3},c_{3}-\min(a,b)\right)\leq c_{0}$,\\\
$c_{0}\leq\min\left(\frac{r-2-b}{2},r-2-a-c_{3},\min(a,b)+c_{3}\right)$,\\\
$\max\left(\frac{b}{2},a-c_{0},c_{0}-\min(a,b)\right)\leq c_{1}$,\\\
$c_{1}\leq\min\left(\frac{r-2-b}{2},r-2-a-c_{0},\min(a,b)+c_{0}\right)$,\\\
$\max\left(\frac{b}{2},a-c_{1},c_{1}-\min(a,b)\right)\leq c_{2}$,\\\
$c_{2}\leq\min\left(\frac{r-2-b}{2},r-2-a-c_{1},\min(a,b)+c_{1}\right)$,\\\
$\max\left(\frac{b}{2},a-c_{2},c_{2}-\min(a,b)\right)\leq c_{3}$,\\\
$c_{3}\leq\min\left(\frac{r-2-b}{2},r-2-a-c_{2},\min(a,b)+c_{2}\right)$\par\par\end{tabular}\right\\}.$

From Figure 16, we can determine the eight tetrahedron terms $|T|_{c}$ and six
edge terms $|\eta|_{c}$ associated to the coloring $c$ which gives us the
following equation:

$\displaystyle
TV_{r,s}(M_{3},\mathcal{T}_{3})=\sum_{(a,b,c_{0},c_{1},c_{2},c_{3})\in\mathcal{A}_{r}(M_{3},\mathcal{T}_{3})}w_{a}w_{b}w_{c_{0}}w_{c_{1}}w_{c_{2}}w_{c_{3}}\cdot$
$\displaystyle\begin{vmatrix}b&a&b\\\
c_{0}&c_{0}&c_{3}\end{vmatrix}\begin{vmatrix}b&a&b\\\
c_{1}&c_{0}&c_{0}\end{vmatrix}$ $\displaystyle\cdot$
$\displaystyle\begin{vmatrix}b&a&b\\\
c_{1}&c_{1}&c_{0}\end{vmatrix}\begin{vmatrix}b&a&b\\\
c_{2}&c_{1}&c_{1}\end{vmatrix}$ $\displaystyle\cdot$
$\displaystyle\begin{vmatrix}b&a&b\\\
c_{2}&c_{2}&c_{1}\end{vmatrix}\begin{vmatrix}b&a&b\\\
c_{3}&c_{2}&c_{2}\end{vmatrix}$ $\displaystyle\cdot$
$\displaystyle\begin{vmatrix}b&a&b\\\
c_{3}&c_{3}&c_{2}\end{vmatrix}\begin{vmatrix}b&a&b\\\
c_{0}&c_{3}&c_{3}\end{vmatrix}.$

Using Proposition 3.6 and a bit of reordering, we can rewrite the summation
as:

$\displaystyle
TV_{r,s}(M_{3},\mathcal{T}_{3})=\sum_{(a,b,c_{0},c_{1},c_{2},c_{3})\in\mathcal{A}_{r}(M_{3},\mathcal{T}_{3})}w_{a}w_{b}w_{c_{0}}w_{c_{1}}w_{c_{2}}w_{c_{3}}\cdot$
$\displaystyle\begin{vmatrix}a&b&b\\\
c_{0}&c_{0}&c_{3}\end{vmatrix}\begin{vmatrix}a&b&b\\\
c_{0}&c_{0}&c_{1}\end{vmatrix}$ $\displaystyle\cdot$
$\displaystyle\begin{vmatrix}a&b&b\\\
c_{1}&c_{1}&c_{0}\end{vmatrix}\begin{vmatrix}a&b&b\\\
c_{1}&c_{1}&c_{2}\end{vmatrix}$ $\displaystyle\cdot$
$\displaystyle\begin{vmatrix}a&b&b\\\
c_{2}&c_{2}&c_{1}\end{vmatrix}\begin{vmatrix}a&b&b\\\
c_{2}&c_{2}&c_{3}\end{vmatrix}$ $\displaystyle\cdot$
$\displaystyle\begin{vmatrix}a&b&b\\\
c_{3}&c_{3}&c_{2}\end{vmatrix}\begin{vmatrix}a&b&b\\\
c_{3}&c_{3}&c_{0}\end{vmatrix}.$

####  6.2.4. Numerical check of the volume conjecture

$0$$10$$20$$30$$40$$12$$14$$16$$18$$r$$\mathcal{R}\left(QV_{r,2}(M_{3})\right)$Vol$(M_{3})=18.038105...$$17.71-12.28\frac{\ln(r-2)}{r-2}-5.09\frac{1}{r-2}$
Figure 17. Graph of the values of $\mathcal{R}\left(QV_{r,2}(M_{3})\right)$
(for $5\leqslant r\leqslant 39$, red triangles) compared with the hyperbolic
volume $\mathrm{Vol}(M_{3})$ (red dashed line) and an interpolation of
$\mathcal{R}\left(QV_{r,2}(M_{3})\right)$ for $r\leqslant 31$ (red curve).

Our conclusions and discussions are almost the same as for the example of
$M_{2}$ in Section 6.1.4. The only differences are that the strange behavior
starts appearing earlier, at $r=31$ (which makes sense since there are more
tetrahedra and thus more terms in the sums), and the constant term from the
interpolating function is equal to the volume $\mathrm{Vol}(M_{3})$ up to
$1.8\%$.

###  6.3. The cases of $M_{4}$ to $M_{7}$

For $4\leqslant g\leqslant 7$, we do not observe a strange behavior of
$\mathcal{R}\left(QV_{r,2}(M_{g})\right)$ as $r$ increases, and we can thus
interpolate all available values with the model
$a+b\cdot\frac{2\pi\ln(r-2)}{r-2}+c\frac{1}{r-2}$. The values of $a,b,c$ are
listed in Figure 11; all computed values of
$\mathcal{R}\left(QV_{r,2}(M_{g})\right)$ and the associated interpolating
functions are displayed in Figure 18.

$10$$20$$30$$40$$10$$20$$30$$r$$\mathcal{R}\left(QV_{r,2}(M_{2})\right)$$11.86-0.83\frac{2\pi\ln(r-2)}{r-2}-5.31\frac{1}{r-2}$
$\mathcal{R}\left(QV_{r,2}(M_{3})\right)$$17.71-1.95\frac{2\pi\ln(r-2)}{r-2}-5.09\frac{1}{r-2}$
$\mathcal{R}\left(QV_{r,2}(M_{4})\right)$$22.91-2.65\frac{2\pi\ln(r-2)}{r-2}-6.74\frac{1}{r-2}$
$\mathcal{R}\left(QV_{r,2}(M_{5})\right)$$27.83-3.23\frac{2\pi\ln(r-2)}{r-2}-8.35\frac{1}{r-2}$
$\mathcal{R}\left(QV_{r,2}(M_{6})\right)$$32.73-3.85\frac{2\pi\ln(r-2)}{r-2}-9.69\frac{1}{r-2}$
$\mathcal{R}\left(QV_{r,2}(M_{7})\right)$$37.25-4.15\frac{2\pi\ln(r-2)}{r-2}-12.12\frac{1}{r-2}$
Figure 18. Graphs of the values of $\mathcal{R}\left(QV_{r,2}(M_{g})\right)$
in function of $r\geqslant 5$ (for $2\leqslant g\leqslant 7$), compared with
their respective best interpolations in the model
$a+b\cdot\frac{2\pi\ln(r-2)}{r-2}+c\frac{1}{r-2}$.

We observe good fits to the model
$a+b\cdot\frac{2\pi\ln(r-2)}{r-2}+c\frac{1}{r-2}$, with $a$ equal to the
expected hyperbolic volume $\mathrm{Vol}(M_{g})$ up to a few percents (see
Figure 11). We conclude that the manifolds $M_{2},\ldots,M_{7}$ seem to
satisfy Conjectures 1.1 and 1.2 numerically.

###  6.4. Behavior of the coefficient $b(M_{g})$ relative to $g$

Let us now delve into Conjecture 1.3. In this section, we assume that
Conjecture 1.2 holds for the manifolds $M_{2},\ldots,M_{7}$ (which is
suggested numerically by the results of the previous sections). We then study
whether or not the coefficient $b$ grows linearly in $g$.

Since we assume that Conjecture 1.2 holds for the manifolds
$M_{2},\ldots,M_{7}$, we can now fix $a=\mathrm{Vol}(M_{g})$ in the model
$a+b\cdot\frac{2\pi\ln(r-2)}{r-2}+c\frac{1}{r-2}$, and look once again for the
best interpolation.

Using find_fit with the new model
$\mathrm{Vol}(M_{g})+b\cdot\frac{2\pi\ln(r-2)}{r-2}+c\frac{1}{r-2}$ yields
different values for $b,c$ than in Section 5.3. These new values are listed in
Figure 19 and the corresponding interpolating functions are displayed in
Figure 20.

$g$ | $r_{max}$ | $\mathrm{Vol}(M_{g})$ | $b$ | $c$
---|---|---|---|---
2 | 33 | 12.04609204 | -1.07486449 | -4.06269480
3 | 31 | 18.03810545 | -2.36670389 | -2.98774665
4 | 27 | 23.60349490 | -3.47345292 | -2.75451472
5 | 23 | 28.98945539 | -4.50837608 | -2.48549875
6 | 23 | 34.28064479 | -5.55394983 | -1.84727854
7 | 19 | 39.51512785 | -6.43483298 | -2.38715613
Figure 19. Values of the interpolating coefficients $a,b,c$ for the model
$\mathrm{Vol}(M_{g})+b\cdot\frac{2\pi\ln(r-2)}{r-2}+c\frac{1}{r-2}$ for
$\mathcal{R}\left(QV_{r,2}(M_{g})\right)$, with $5\leqslant r\leqslant
r_{max}$.
$10$$20$$30$$40$$10$$20$$30$$r$$\mathcal{R}\left(QV_{r,2}(M_{2})\right)$$\mathrm{Vol}(M_{2})-1.07\frac{2\pi\ln(r-2)}{r-2}-4.06\frac{1}{r-2}$
$\mathcal{R}\left(QV_{r,2}(M_{3})\right)$$\mathrm{Vol}(M_{3})-2.36\frac{2\pi\ln(r-2)}{r-2}-2.98\frac{1}{r-2}$
$\mathcal{R}\left(QV_{r,2}(M_{4})\right)$$\mathrm{Vol}(M_{4})-3.47\frac{2\pi\ln(r-2)}{r-2}-2.75\frac{1}{r-2}$
$\mathcal{R}\left(QV_{r,2}(M_{5})\right)$$\mathrm{Vol}(M_{5})-4.51\frac{2\pi\ln(r-2)}{r-2}-2.48\frac{1}{r-2}$
$\mathcal{R}\left(QV_{r,2}(M_{6})\right)$$\mathrm{Vol}(M_{6})-5.55\frac{2\pi\ln(r-2)}{r-2}-1.84\frac{1}{r-2}$
$\mathcal{R}\left(QV_{r,2}(M_{7})\right)$$\mathrm{Vol}(M_{7})-6.43\frac{2\pi\ln(r-2)}{r-2}-2.38\frac{1}{r-2}$
Figure 20. Graphs of the values of $\mathcal{R}\left(QV_{r,2}(M_{g})\right)$
in function of $r\geqslant 5$ (for $2\leqslant g\leqslant 7$), compared with
their respective best interpolations in the model
$\mathrm{Vol}(M_{g})+b\cdot\frac{2\pi\ln(r-2)}{r-2}+c\frac{1}{r-2}$.

As expected by their definitions, the interpolations of Figure 20 are less
fitting than the ones of Figure 18, but they still seem satisfactory.

Figure 21 displays the values of the coefficient $b$ in function of $g$ (as
black asterisks), with the corresponding best linear interpolation (the green
line). The coefficient of determination of this linear interpolation is
$R^{2}=0.9967$, which gives much credit to the hypothesis of the affine
behaviour of $b$.

$1$$2$$3$$4$$5$$6$$7$$-6$$-4$$-2$$g$$b$ coefficient for
$\mathcal{R}\left(QV_{r,2}(M_{g})\right)$$0.9061-1.068\ g$ Figure 21. Values
of the coefficient $b(M_{g})$ for $2\leqslant g\leqslant 7$ (black asterisks),
and the associated interpolating affine function (green line), with
coefficient of determination $R^{2}=0.9967$.

What precedes thus yields a satisfying numerical check of Conjecture 1.3 for
the family of manifolds $M_{g}$.

More precisely, the interpolating affine function is computed to be

$0.9061-1.068g.$

Of course, more data would give us an interpolating affine function closer to
the expected one, but as the slope $-1.068$ is already quite close to $-1$, it
is not unreasonable to look for a general behavior of $b$ in the form of

$b(M)\overset{?}{=}\text{constant}-\frac{1}{2}\chi(\partial M),$

since in the specific case of Frigerio’s manifolds we have

$b(M_{g})\underset{2\leqslant g\leqslant
7}{\approx}\text{constant}-g=\text{constant}-\frac{1}{2}\chi(\partial M_{g}).$

##  7\. Discussion and further directions

  * •

It would be interesting to understand the origin of the pattern breaks for
$\mathcal{R}\left(QV_{r,2}(M_{2})\right)$ and
$\mathcal{R}\left(QV_{r,2}(M_{3})\right)$. If, as we surmise, they come from
numerical approximations by the machine for terms of different magnitudes,
then this hypothesis could be tested by refining our code and examining the
range of magnitudes of the terms in the sum when $r$ grows larger. The works
of Maria-Rouillé [12] seem like a promising direction to follow.

  * •

Conjecture 1.3 appears to be satisfied for the manifolds $M_{g}$, but it would
be interesting to test it for other families of manifolds with diverse
boundary components. Furthermore, one could try to prove (or disprove!)
rigorously that $b$ has an affine behavior of the form
$-\frac{1}{2}\chi(\partial M)+constant$, via combinatorial arguments on the
triangulations (how the numbers of vertices, edges and tetrahedra are related
to $\chi(\partial M)$) and asymptotics in $r$ of the terms associated to
regular vertices, edges and tetrahedra in the definition of the Turaev–Viro
invariants.

  * •

The extended volume conjectures as stated in Conjectures 1.2 and 1.3 already
(or may possibly) admit variants for other quantum invariants. Can the methods
used in the present paper be applied for these other invariants? The manifolds
$M_{g}$ seem especially convenient to study for invariants defined on
(ordered) triangulations.

## Acknowledgements

The first author was supported by the FNRS in his ”Research Fellow” position
at UCLouvain, under under Grant no. 1B03320F. The second author would like to
thank his two supervisors (Pedro Vaz and the first author) for their
intellectual and emotional support throughout his Master’s thesis that gave
rise to the current paper. Both authors thank Pedro Vaz for his continuous
involvement in the project, and Renaud Detcherry and François Costantino for
helpful discussions.

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