aainstitutetext: Department of Physics, Institute of Science, Banaras Hindu
University,
Varanasi, 221005, Indiabbinstitutetext: Center for Theoretical Physics,
College of Physical Science and Technology, Sichuan University,
Chengdu, 610064, Chinaccinstitutetext: Department of Physics, Indian Institute
of Technology Bombay,
Powai, Mumbai, 400076, India ddinstitutetext: Department of Mathematics,
Indian Institute of Science Education and Research, Pune,
Pashan, Pune, 411008, India

# Topological entanglement and hyperbolic volume

Aditya Dwivedi b , Siddharth Dwivedi a , Bhabani Prasad Mandal c , Pichai
Ramadevi d , Vivek Kumar Singh<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The entanglement entropy of many quantum systems is difficult to compute in
general. They are obtained as a limiting case of the Rényi entropy of index
$m$, which captures the higher moments of the reduced density matrix. In this
work, we study pure bipartite states associated with $S^{3}$ complements of a
two-component link which is a connected sum of a knot $\mathcal{K}$ and the
Hopf link. For this class of links, the Chern-Simons theory provides the
necessary setting to visualise the $m$-moment of the reduced density matrix as
a three-manifold invariant $Z(M_{\mathcal{K}_{m}})$, which is the partition
function of $M_{\mathcal{K}_{m}}$. Here $M_{\mathcal{K}_{m}}$ is a closed
3-manifold associated with the knot $\mathcal{K}_{m}$, where $\mathcal{K}_{m}$
is a connected sum of $m$-copies of $\mathcal{K}$ (i.e.,
$\mathcal{K}\\#\mathcal{K}\ldots\\#\mathcal{K}$) which mimics the well-known
replica method. We analyse the partition functions $Z(M_{\mathcal{K}_{m}})$
for SU(2) and SO(3) gauge groups, in the limit of the large Chern-Simons
coupling $k$. For SU(2) group, we show that $Z(M_{\mathcal{K}_{m}})$ can grow
at most polynomially in $k$. On the contrary, we conjecture that
$Z(M_{\mathcal{K}_{m}})$ for SO(3) group shows an exponential growth in $k$,
where the leading term of $\ln Z(M_{\mathcal{K}_{m}})$ is the hyperbolic
volume of the knot complement $S^{3}\backslash\mathcal{K}_{m}$. We further
propose that the Rényi entropies associated with SO(3) group converge to a
finite value in the large $k$ limit. We present some examples to validate our
conjecture and proposal.

##  1 Introduction

The study of quantum entanglement and finding the possible patterns of
entanglement that can emerge in a quantum field theory (QFT) is generally an
important question in quantum mechanics and quantum information theory.
However, due to large degrees of freedom, it is a difficult exercise to
analyse the entanglement structures in a generic QFT. Nevertheless, a class of
QFTs known as ‘topological quantum field theories’ (TQFT’s) provides a
tractable system to analyse entanglement structures.

The three-dimensional Chern-Simons theory is one such TQFT Witten:1988hf
which provides a natural framework to study invariants of knots, links, and
three-manifolds. Interestingly, the Chern-Simons path integral on a three-
manifold $M$ with boundary $\partial M=\Sigma$ is given by a state belonging
to a finite-dimensional Hilbert space $\mathcal{H}_{\Sigma}$. Such a finite-
dimensional Hilbert space is a key behind explicitly computing the
entanglement measures.

There are two different topological set-ups to obtain the entanglement entropy
in Chern-Simons theory as shown in figure 1. In figure 1($a$), the manifold
has a single torus boundary which is bi-partitioned into spatially connected
regions $A$ and its complement $A^{c}$.

Figure 1: The two approaches to study the topological entanglement: the
manifold in ($a$) has a single torus boundary, which is bi-partitioned into
spatially connected regions $A$ and $A^{c}$. The manifold in ($b$) has two
disjoint torus boundaries.

When we trace out the region $A^{c}$, we obtain the reduced density matrix
$\rho_{A}$ and the corresponding ‘topological entanglement entropy’. Such an
entropy is independent of the length or the area of the region $A$ or $A^{c}$
and have been studied in Kitaev:2005dm ; Levin:2006zz ; Dong:2008ft . Another
approach to study the topological entanglement was given in
Balasubramanian:2016sro , where the three-manifold is actually a link
complement111Given a link $\mathcal{L}$ embedded in $S^{3}$, the link
complement $S^{3}\backslash\mathcal{L}$ is a three-dimensional manifold which
is obtained by removing a tubular neighborhood around $\mathcal{L}$ from
$S^{3}$, i.e $S^{3}\backslash\mathcal{L}\equiv
S^{3}-\text{interior}(\mathcal{L}_{\text{tub}})$. of $S^{3}$ with two or more
disjoint torus boundaries as shown in figure 1($b$). The topological
entanglement structure can be obtained by tracing out one of the boundary
components, which is often termed as ‘multi-boundary entanglement’. We refer
the interested readers to Balasubramanian:2016sro ; Dwivedi:2017rnj ;
Balasubramanian:2018por ; Hung:2018rhg ; Melnikov:2018zfn ; Camilo:2019bbl ;
Dwivedi:2019bzh ; Buican:2019evc ; Zhou:2019ezk ; Dwivedi:2020jyx ;
Dwivedi:2020rlo  for the recent developments in this study. For the three-
manifold $M$ whose boundary consists of multiple disjoint torus boundaries
($\partial M=\Sigma_{1}\sqcup\Sigma_{2}\sqcup\ldots\sqcup\Sigma_{n}$), the
associated Hilbert space is

$\mathcal{H}_{\partial
M}=\mathcal{H}_{\Sigma_{1}}\otimes\mathcal{H}_{\Sigma_{2}}\otimes\ldots\otimes\mathcal{H}_{\Sigma_{n}}~{}.$
(1)

When $M$ is a link complement ($M=S^{3}\backslash\mathcal{L}$), the
probability amplitudes of the associated state $\ket{\mathcal{L}}$ are the
Chern-Simons partition functions $Z(S^{3};\mathcal{L})$ of $S^{3}$ in the
presence of the link $\mathcal{L}$ (see Balasubramanian:2016sro ), which are
proportional to the link invariants Witten:1988hf . Therefore, the
entanglement measures for such a state can be written in terms of the link
invariants of $\mathcal{L}$.

In this work, we analyse the semiclassical (large $k$) asymptotics of the
trace of the unnormalised reduced density matrices
($\text{Tr}[\sigma(\mathcal{L})]$) of two-party states associated with the
link complement $S^{3}\backslash\mathcal{L}$ for a class of two-component
links $\mathcal{L}$ viewed as connected sum of prime knots $\mathcal{K}$ with
the Hopf link $T_{2,2}$:

$\mathcal{L}=\mathcal{K}\\#T_{2,2}~{}.$ (2)

Qualitatively, $\text{Tr}[\sigma(\mathcal{K}\\#T_{2,2})]$ describes the Chern-
Simons partition function of some closed three-manifold $M_{\mathcal{K}}$
(although the topology of $M_{\mathcal{K}}$ is not known to us)
Melnikov:2018zfn . For the class of links (2), this trace is given as
Balasubramanian:2016sro :

$\text{Tr}[\sigma(\mathcal{K}\\#T_{2,2})]=Z(M_{\mathcal{K}})=\sum_{R}\absolutevalue{H_{R}(\mathcal{K};q=e^{\frac{2\pi
i}{k+y}})}^{2}~{},$ (3)

where $H_{R}(\mathcal{K};q)$ are the reduced quantum invariants of knot
$\mathcal{K}$ with $y$ being the dual Coxeter number of the gauge group (see
the details in section 3). Such a trace (3) for SU(2) gauge group almost
resembles the Turaev-Viro invariant
$\text{TV}_{r}(S^{3}\backslash\mathcal{K}\,;t=e^{\frac{\pi i}{r}})$ for the
knot complement $S^{3}\backslash\mathcal{K}$ with $t^{2}=q$ and $r=(k+2)$
zbMATH00167790 ; detcherry2018turaev . In fact, this resemblance to
$\text{TV}_{r}(S^{3}\backslash\mathcal{K}\,;t)$ motivates us to investigate
the large $k$ asymptotics of (3) for the class of links
$\mathcal{K}\\#T_{2,2}$ within the Chern-Simons theory.

For SU(2) gauge group, we observe the trace (3) follows the same polynomial
growth in $k$ as that of Turaev-Viro invariant
$\text{TV}_{r}(S^{3}\backslash\mathcal{K}\,;t=e^{\frac{\pi i}{r}})$ with
$r=(k+2)$. Incidentally, the large $k$ asymptotics of Turaev-Viro invariant
detcherry2018turaev  evaluated at a different root of unity $t=e^{\frac{2\pi
i}{r^{\prime}}}$, with $r^{\prime}=(k+1)$ being an odd integer, captures the
hyperbolic volume of the knot complement $S^{3}\backslash\mathcal{K}$. Such a
change of variable naturally happens when we study the knot invariants in
SO(3) Chern-Simons theory (we refer to section 3 for more details).

Hence our focus in this work is to investigate the trace (3) for the links of
type $\mathcal{K}\\#T_{2,2}$ within the context of SO(3) Chern-Simons theory.
Interestingly, our numerical analysis, combined with the Kashaev’s conjecture
kashaev1997hyperbolic , is instrumental in extracting the geometrical features
of $S^{3}\backslash\mathcal{K}$ from (3). Specifically, we conjecture an
exponential growth of
$\text{Tr}[\sigma(\mathcal{K}\\#T_{2,2})]=Z(M_{\mathcal{K}})$ with $k$, where
the growth rate is determined by the hyperbolic volume of
$S^{3}\backslash\mathcal{K}$.

We further propose that the Rényi entropies for SO(3) gauge group associated
with the state $\ket{\mathcal{K}\\#T_{2,2}}$ converge to a finite value as
$k\to\infty$, although currently we do not have a geometric or topological
interpretation of this limit.

The paper is organized as follows. In section 2, we discuss the preliminaries,
including a brief discussion of our set-up. In section 3, we begin by
analysing the reduced density matrices and their traces associated with the
links of type $\mathcal{K}\\#T_{2,2}$. We further analyse their large $k$
asymptotics for two gauge groups: SU(2) and SO(3), respectively. In section 4,
we present several examples where we do explicit numerical computations to
verify the results conjectured and proposed in section 3 for the SO(3) group.
We finally conclude in section 5.

##  2 Preliminaries

###  2.1 Chern-Simons theory and multi-boundary states

Three-dimensional Chern-Simons theory based on a compact gauge group $G$ and
coupling $k\in\mathbb{Z}$ is described by the following metric independent
action:

$S(A)=\frac{k}{4\pi}\int_{M}\text{Tr}\left(A\wedge dA+\frac{2}{3}A\wedge
A\wedge A\right)~{}.$ (4)

Here $A=A_{\mu}dx^{\mu}$ is a matrix-valued gauge field, and $M$ denotes a
three-manifold. The gauge-invariant operators are the Wilson loops. Given a
knot $\mathcal{K}$ embedded in $M$, the Wilson loop operator is defined by
taking the trace of the holonomy of $A$ around $\mathcal{K}$ :

$W_{R}(\mathcal{K})=\text{Tr}_{R}\,P\exp\left(i\oint_{\mathcal{K}}A\right)~{}.$
(5)

Note that the knot $\mathcal{K}$ carries a representation $R$ of the gauge
group $G$, and the trace is over the matrix corresponding to the
representation $R$. Chern-Simons partition function, which encodes the
topological information of the three-manifold $M$, is given by

$Z(M)=\int e^{iS(A)}\mathcal{D}A~{},$ (6)

where $\mathcal{D}A$ denotes integration over all the gauge-invariant classes
of connections. In fact, $Z(M)$ is referred to as a three-manifold invariant
(up to an overall normalisation) in the literature.

The partition function of $M$ in the presence of knots and links is obtained
by inserting the appropriate Wilson loop operators in the integral. For
example, consider a link $\mathcal{L}$ made of component knots
$\mathcal{K}_{1}$, $\mathcal{K}_{2}$ given by Wilson link operator
$W_{R1,R_{2}}(\mathcal{L})=W_{R_{1}}(\mathcal{K}_{1})W_{R_{2}}(\mathcal{K}_{2})$.
Note that the representations $R_{1},R_{2}$ are placed on the component knots
$\mathcal{K}_{1},\mathcal{K}_{2}$ respectively. The partition function in the
presence of such a link is given by

$Z(M;\mathcal{L}[R_{1},R_{2}])=\int
e^{iS(A)}\,W_{R_{1}}(\mathcal{K}_{1})W_{R_{2}}(\mathcal{K}_{2})\,\mathcal{D}A~{}.$
(7)

Suppose the manifold is closed with no boundary, the link invariants
$V_{R_{1},R_{2}}(\mathcal{L})$ with the two components carrying representation
$R_{1},R_{2}$ are given by the expectation value of the Wilson link operator:

$V_{R_{1},R_{2}}(\mathcal{L})=\frac{1}{Z(M)}\int
e^{iS(A)}\,W_{R_{1}}(\mathcal{K}_{1})W_{R_{2}}(\mathcal{K}_{2})\,\mathcal{D}A~{}.$
(8)

When $M$ has a boundary $\Sigma$, the path integral of the theory on $M$ with
the Wilson link $\mathcal{L}$ insertion is interpreted as a state
$\ket{\Psi}\equiv\ket{\mathcal{L}}\in\mathcal{H}_{\Sigma}$. Further note that
if we reverse the orientation of the boundary, the associated Hilbert space
becomes the dual of the original Hilbert space:

$\mathcal{H}_{\Sigma^{*}}=\mathcal{H}^{*}_{\Sigma}~{}.$ (9)

As a result, there exists a natural pairing, the inner product
$\langle\Phi|\Psi\rangle$ for any two states
$\ket{\Psi}\in\mathcal{H}_{\Sigma}$ and
$\bra{\Phi}\in\mathcal{H}_{\Sigma^{*}}$. This technique can be used to compute
the partition functions $Z(M)$ of complicated closed manifolds by gluing two
disconnected pieces along their oppositely oriented boundary as shown in the
figure 2.

Figure 2: Two manifolds on left with same boundary but opposite orientation.
The path integral on these manifolds gives states
$\bra{\phi}\in\mathcal{H}_{\Sigma^{*}}$ and
$\ket{\psi}\in\mathcal{H}_{\Sigma}$. The inner product $\bra{\phi}\ket{\psi}$
will be the partition function of a closed three-manifold (shown in right)
obtained by gluing.

When the boundary of the manifold $M$ consists of disjoint components, i.e.,
$\Sigma=\Sigma_{1}\sqcup\Sigma_{2}$, the Hilbert space associated with
$\Sigma$ is the tensor product of Hilbert spaces associated with each
component, i.e.

$\mathcal{H}_{\Sigma}=\mathcal{H}_{\Sigma_{1}}\otimes\mathcal{H}_{\Sigma_{2}}~{}.$
(10)

For the states $\ket{\Psi}\in\mathcal{H}_{\Sigma}$, we can study the
entanglement structure by tracing out the Hilbert space
$\mathcal{H}_{\Sigma_{1}}~{}{\rm or}~{}\mathcal{H}_{\Sigma_{2}}$. In this
work, we will consider the link complement manifold

$M=S^{3}\backslash\mathcal{L}~{},$ (11)

whose boundary is $\Sigma=T^{2}\sqcup T^{2}$ for any two-component link
$\mathcal{L}$. We denote the state associated with
$S^{3}\backslash\mathcal{L}$ as
$\ket{\mathcal{L}}\in\mathcal{H}_{T^{2}}\times\mathcal{H}_{T^{2}}$. We will
now review the essential steps Balasubramanian:2016sro  of computing such
states. As discussed in Ref.Witten:1988hf , the basis states of the Hilbert
space $\mathcal{H}_{T^{2}}$ are in one-to-one correspondence with the
integrable representations of the affine Lie algebra $\hat{\mathfrak{g}}_{k}$
at level $k$. Here $\mathfrak{g}$ represents the Lie algebra associated with
the group $G$. The Chern-Simons path integral on a solid torus with a Wilson
loop along the non-contractible cycle carrying integrable representation
$\alpha$ is given by the basis state $\ket{e_{\alpha}}$:

$\begin{array}[]{c}\includegraphics[width=108.405pt]{basis.png}\end{array}=\ket{e_{\alpha}}~{}.$
(12)

Using such basis states, we can expand the bipartite state $\ket{\mathcal{L}}$
as:

$\ket{\mathcal{L}}=\sum_{\alpha}\sum_{\beta}C_{\alpha\beta}\ket{e_{\alpha}}\otimes\ket{e_{\beta}}=\sum_{\alpha}\sum_{\beta}C_{\alpha\beta}\ket{e_{\alpha},e_{\beta}}~{},$
(13)

where $C_{\alpha\beta}$ are the complex coefficients given by partition
function (7) Balasubramanian:2016sro :

$C_{\alpha\beta}=\bra{e_{\alpha},e_{\beta}}\ket{\mathcal{L}}=Z(S^{3};\mathcal{L}[\alpha,\beta])~{}.$
(14)

Hence the two-component link state can be written as

$\ket{\mathcal{L}}=\sum_{\alpha}\sum_{\beta}Z(S^{3};\mathcal{L}[\alpha,\beta])\ket{e_{\alpha},e_{\beta}}~{}.$
(15)

To study its entanglement properties, we require a reduced density matrix by
partial tracing one of the two Hilbert spaces. We will now present these
entanglement features in the following subsection.

###  2.2 Reduced density matrix and Rényi entropy

Given the link state
$\ket{\mathcal{L}}\in\mathcal{H}_{1}\otimes\mathcal{H}_{2}$ in (15), the
corresponding density matrix operator is

$\hat{\rho}(\mathcal{L})=\ket{\mathcal{L}}\bra{\mathcal{L}}~{}.$ (16)

The reduced density matrix is computed by tracing out one of the Hilbert
space, say $\mathcal{H}_{2}$:

$\rho(\mathcal{L})=\frac{\text{Tr}_{\mathcal{H}_{2}}[\hat{\rho}(\mathcal{L})]}{\innerproduct{\mathcal{L}}{\mathcal{L}}}=\frac{1}{\innerproduct{\mathcal{L}}{\mathcal{L}}}\sum_{\alpha}\expectationvalue{\hat{\rho}(\mathcal{L})}{e_{\alpha}}~{},$
(17)

where the factor $\innerproduct{\mathcal{L}}{\mathcal{L}}$ in the denominator
ensures that the trace of $\rho(\mathcal{L})$ is unity. For convenience, we
will work with the unnormalised version of the reduced density matrix as
follows:

$\sigma(\mathcal{L})=\text{Tr}_{\mathcal{H}_{2}}[\hat{\rho}(\mathcal{L})]~{}.$
(18)

The entanglement measures associated with $\ket{\mathcal{L}}$ can be computed
from the matrix $\sigma(\mathcal{L})$. For example, the Rényi entropy of index
$m$ can be written in terms of $m$-moments or the trace of the $m^{\text{th}}$
power of $\sigma(\mathcal{L})$ as following:

$\mathcal{R}_{m}=\frac{1}{1-m}\ln\left(\frac{\text{Tr}[\sigma^{m}(\mathcal{L})]}{(\text{Tr}[\sigma(\mathcal{L})])^{m}}\right)~{}.$
(19)

The entanglement entropy is given as,

$\mathcal{E}=\lim_{m\to 1}\mathcal{R}_{m}~{}.$ (20)

There also exists a minimum entropy which is controlled by the maximum
eigenvalue of the reduced density matrix. Further, in the large $m$ limit, the
corresponding Rényi entropy is minimum:

$\mathcal{R}_{\text{min}}\equiv\mathcal{R}_{\infty}=\lim_{m\to\infty}\mathcal{R}_{m}~{}.$
(21)

In the next section, we will study some of the properties of the density
matrices and the entanglement measures associated with a class of two-
component links $\mathcal{L}$ which is a connected sum of a prime knot
$\mathcal{K}$ with Hopf link.

##  3 Bi-partite state: Connected sum of a knot $\mathcal{K}$ and Hopf link

In this work, we consider the two-component link
$\mathcal{L}=\mathcal{K}\\#T_{2,2}$ which is a connected sum of a prime knot
$\mathcal{K}$ and the Hopf link $T_{2,2}$. A typical example of such a link is
shown in the figure 3 when $\mathcal{K}$ is the figure-eight knot.

Figure 3: The two component link $4_{1}\\#T_{2,2}$ which is the connected sum
of figure-eight knot and Hopf link.

In this section, we will closely analyse the bi-partite state
$\ket{\mathcal{K}\\#T_{2,2}}$ and its properties. In particular, our focus
will be on the traces of the unnormalised reduced density matrices. As
mentioned in the introduction, we will see that these traces have a close
resemblance to the Turaev-Viro invariant zbMATH00167790 ; detcherry2018turaev
, motivating us to focus on the Chern-Simons theory based on SU(2) as well as
SO(3) gauge groups.

We will first start with a generic discussion on the density matrices
associated with $\ket{\mathcal{K}\\#T_{2,2}}$ in the following subsection.

###  3.1 Reduced density matrices

Following the prescription given in the previous section, we can write the
state associated with the link $\mathcal{K}\\#T_{2,2}$ as

$\ket{\mathcal{K}\\#T_{2,2}}=\sum_{\alpha,\,\beta}Z(S^{3},\mathcal{K}_{\alpha}\,\\#\,T_{2,2}[\alpha,\beta])\ket{e_{\alpha},e_{\beta}}~{},$
(22)

where the probability amplitudes are the Chern-Simons partition functions of
$S^{3}$ in presence of the link $\mathcal{K}\,\\#\,T_{2,2}$. The labels
$\alpha$ and $\beta$ are the representations carried by $\mathcal{K}$ and
unknot components of the link respectively (as shown in the example in figure
3). This partition function can be computed following Ref.Witten:1988hf  and
is given as,

$Z(S^{3},\mathcal{K}_{\alpha}\,\\#\,T_{2,2}[\alpha,\beta])=\frac{Z(S^{3},\mathcal{K}_{\alpha})\,Z(S^{3},T_{2,2}[\alpha,\beta])}{Z(S^{3},U_{\alpha})}=Z(S^{3},\mathcal{K}_{\alpha})\frac{\mathcal{S}_{\alpha\beta}}{\mathcal{S}_{0\alpha}}~{},$
(23)

where $U_{\alpha}$ is the unknot with representation $\alpha$. In the last
equality, the partition functions of Hopf link and unknot are given in terms
of the elements of the modular transformation matrix222The operator
$\mathcal{S}$ is one of the generators of the modular group SL(2,
$\mathbb{Z}$). In the above discussion, the operator $\mathcal{S}$ is written
in a matrix form with element $\mathcal{S}_{\alpha\beta}$, where $\alpha$ and
$\beta$ label the integrable representations of affine Lie algebra
$\mathfrak{g}_{k}$ at level $k$. For $\mathfrak{su}(2)_{k}$ and
$\mathfrak{so}(3)_{k}$ algebras, the matrix $\mathcal{S}$ is symmetric,
unitary and satisfies $\mathcal{S}^{*}=\mathcal{S}$. $\mathcal{S}$. Thus we
can write the state as:

$\ket{\mathcal{K}\\#T_{2,2}}=\sum_{\alpha,\,\beta}Z(S^{3},\mathcal{K}_{\alpha})\frac{\mathcal{S}_{\alpha\beta}}{\mathcal{S}_{0\alpha}}\,\ket{e_{\alpha},e_{\beta}}~{}.$
(24)

As the entanglement properties of a state do not change under a local unitary
change of the basis, we simplify the above state by performing the following
transformation:

$\ket{e_{\beta}}=\sum_{x}\mathcal{S}_{\beta x}^{*}\,\ket{e_{x}}~{}.$ (25)

Substituting such a transformation in (24) and using the unitary property of
the $\mathcal{S}$ matrix, we get

$\ket{\mathcal{K}\\#T_{2,2}}=\sum_{\alpha}\frac{Z(S^{3},\mathcal{K}_{\alpha})}{\mathcal{S}_{0\alpha}}\,\ket{e_{\alpha},e_{\alpha}}~{}.$
(26)

Interestingly, the probability amplitudes appearing in the above state are
precisely the reduced quantum invariants333The reduced invariants are
normalised such that the unknot invariant is unity, i.e. $H_{\alpha}(U;q)=1$.
of the knot $\mathcal{K}$ evaluated at a specific root of unity Witten:1988hf
:

$\frac{Z(S^{3},\mathcal{K}_{\alpha})}{\mathcal{S}_{0\alpha}}=\frac{Z(S^{3},\mathcal{K}_{\alpha})/Z(S^{3})}{\mathcal{S}_{0\alpha}/Z(S^{3})}=\frac{V_{\alpha}(\mathcal{K})}{V_{\alpha}(U)}=H_{\alpha}(\mathcal{K};q=e^{\frac{2\pi
i}{k+y}})~{},$ (27)

where $H_{\alpha}(\mathcal{K};q)$ are the reduced invariants of knot
$\mathcal{K}$ evaluated at $q=e^{\frac{2\pi i}{k+y}}$ with $y$ being the dual
Coxeter number of the gauge group. Thus we arrive at the following state:

$\ket{\mathcal{K}\\#T_{2,2}}=\sum_{\alpha}H_{\alpha}(\mathcal{K};q=e^{\frac{2\pi
i}{k+y}})\,\ket{e_{\alpha},e_{\alpha}}~{}.$ (28)

The unnormalised reduced density matrix (18) for this state is a diagonal
matrix:

$\sigma(\mathcal{K}\\#T_{2,2})=\text{diag}\left\\{\absolutevalue{H_{\alpha}(\mathcal{K};q=e^{\frac{2\pi
i}{k+y}})}^{2}\quad;\quad\alpha\in\text{Integrable
representations}\right\\}~{},$ (29)

where the entries are labeled by various integrable representations $\alpha$.
The $m$-moment or the trace of the $m^{\text{th}}$ power of this matrix is:

$\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]=\sum_{\alpha}\absolutevalue{H_{\alpha}(\mathcal{K};q=e^{\frac{2\pi
i}{k+y}})}^{2m}~{}.$ (30)

We also know that the trace of the reduced density matrix
$\sigma(\mathcal{K}\\#T_{2,2})$ can be viewed as gluing oppositely oriented
torus boundaries resulting in a closed three-manifold without any boundary.
Some examples have been qualitatively discussed in Ref.Melnikov:2018zfn . For
clarity, let us look at the simplest link $\mathcal{L}=U\\#T_{2,2}$ involving
the unknot $U$ whose Jones polynomial is 1. Clearly,

$\text{Tr}[\sigma(U\\#T_{2,2})]=\sum_{\alpha}(1)={\rm
dim}~{}\mathcal{H}_{T^{2}}=Z(T^{2}\times S^{1})~{},$ (31)

giving the Chern-Simons partition function of $M_{U}\equiv T^{2}\times S^{1}$.
Replacing $U$ by a non-trivial prime knot $\mathcal{K}$ would eventually give
us:

$\text{Tr}[\sigma(\mathcal{K}\\#T_{2,2})]=\sum_{\alpha}\absolutevalue{H_{\alpha}(\mathcal{K};q=e^{\frac{2\pi
i}{k+y}})}^{2}=Z(M_{\mathcal{K}})$ (32)

for some closed three-manifold $M_{\mathcal{K}}$, though we cannot specify the
explicit topology of $M_{\mathcal{K}}$. Further, the $m$-moment of the
unnormalised reduced density matrix $\sigma(\mathcal{K}\\#T_{2,2})$ can be
viewed as joining $m$-copies of the oppositely oriented $T^{2}$ boundaries.
This is called the replica method Calabrese:2004eu  and it again results in a
closed three-manifold. In the following subsection, we will present the knot
theoretic interpretation of the replica method and their relations to
$m$-moments of $\sigma(\mathcal{K}\\#T_{2,2})$ defining three-manifold
invariants. We will show that the Rényi entropies of index $m$ can be written
in terms of these three-manifold invariants.

###  3.2 Rényi entropies and three-manifold invariants

The Rényi entropy of index $m$ associated with the reduced density matrix of
state $\ket{\mathcal{K}\\#T_{2,2}}$ can be computed as:

$\mathcal{R}_{m}=\frac{1}{1-m}\ln\left[\frac{\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]}{(\text{Tr}[\sigma(\mathcal{K}\\#T_{2,2})])^{m}}\right]=\frac{1}{1-m}\ln\left[\frac{\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]}{Z(M_{\mathcal{K}})^{m}}\right]~{}.$
(33)

We aim to write the numerator of the logarithmic term also as a three-manifold
invariant. Just like our discussion that
$\text{Tr}[\sigma(\mathcal{K}\\#T_{2,2})]=Z(M_{\mathcal{K}})$, a similar
topological attribution is also true for the higher moments of the reduced
density matrix. We will now elaborate the knot theoretic picture of the
replica trick to obtain $\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]$.

Replica method involving $m$-copies is achievable by replacing $\mathcal{K}$
by the following connected sum of knots:

$\mathcal{K}_{m}\equiv\underbrace{\mathcal{K}\,\\#\,\mathcal{K}\,\\#\,\ldots\\#\,\mathcal{K}}_{m}~{}.$
(34)

Incidentally, the reduced knot invariants for connected sum of two knots
$\mathcal{K}_{1}\\#\mathcal{K}_{2}$ is equal to the product of the reduced
knot invariants for knots $\mathcal{K}_{1}$ and $\mathcal{K}_{2}$. Using this
fact and (30), we will have:

$\text{Tr}[\sigma(\mathcal{K}_{m}\\#T_{2,2})]=\sum_{\alpha}\absolutevalue{H_{\alpha}(\mathcal{K}_{m};q)}^{2}=\sum_{\alpha}\absolutevalue{H_{\alpha}(\mathcal{K};q)}^{2m}=\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]~{},$
(35)

with $q=e^{\frac{2\pi i}{k+y}}$. Thus using (32) in (35), we realize:

$\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]=Z(M_{\mathcal{K}_{m}})$ (36)

and hence we can write the Rényi entropies in terms of three-manifold
invariants:

$\boxed{\mathcal{R}_{m}=\frac{1}{1-m}\ln\left[\frac{Z(M_{\mathcal{K}_{m}})}{Z(M_{\mathcal{K}})^{m}}\right]}~{}.$
(37)

So far, our discussion has been for a generic gauge group. In the following
subsections, we consider the cases when the gauge group is SU(2) and SO(3,
respectively. In particular, we focus on the large $k$ asymptotics of the
traces $\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]=Z(M_{\mathcal{K}_{m}})$,
which almost resembles the Turaev-Viro invariants evaluated at certain roots
of unity detcherry2018turaev . We start with the SU(2) case in the following
subsection.

###  3.3 SU(2) gauge group & $m$-moments of density matrix

The integrable representations for the affine algebra $\mathfrak{su}(2)_{k}$
are given as:

$\left\\{\,\underbrace{\tiny\hbox{}{\vbox{\offinterlineskip\vbox{\hrule
height=0.3pt\hbox{\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt} \hrule
height=0.3pt}\vspace{-\y@linethick} }}}_{\alpha}\,\,:\,\,0\leq\alpha\leq
k\right\\}~{}.$ (38)

Hence the basis states of $\mathcal{H}_{T^{2}}$ can be accordingly labeled:

$\text{basis}(\mathcal{H}_{T^{2}})=\left\\{\ket{e_{\alpha}}\,\,:\,\,\alpha\equiv\underbrace{\tiny\hbox{}{\vbox{\offinterlineskip\vbox{\hrule
height=0.3pt\hbox{\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt} \hrule
height=0.3pt}\vspace{-\y@linethick} }}}_{\alpha}\right\\}~{}.$ (39)

Using (28) and noting that the reduced invariants $H_{\alpha}$ are the colored
Jones polynomials, the state is written as:

$\ket{\mathcal{K}\\#T_{2,2}}=\sum_{\alpha=0}^{k}J_{\alpha}(\mathcal{K}\,;e^{\frac{2\pi
i}{k+2}})\,\ket{e_{\alpha},e_{\alpha}}$ (40)

and the $m$-moment or the trace of the $m^{\text{th}}$ power of the
unnormalised reduced density matrix associated with
$\ket{\mathcal{K}\\#T_{2,2}}$ will be:

$\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]=\sum_{\alpha=0}^{k}\absolutevalue{J_{\alpha}(\mathcal{K}\,;e^{\frac{2\pi
i}{k+2}})}^{2m}~{}.$ (41)

We will now show that the large $k$ asymptotics of (41) is the same as that of
Turaev-Viro invariant zbMATH00167790  of $S^{3}\backslash\mathcal{K}_{m}$ at a
specific root of unity.

####  3.3.1 Large $k$ asymptotics: The polynomial growth

Following detcherry2018turaev , the Turaev-Viro invariant of
$S^{3}\backslash\mathcal{K}_{m}$ evaluated at $e^{\frac{\pi i}{k+2}}$ with
integer $k\geq 1$ can be written in terms of values of its colored Jones
polynomials:444The colored Jones $J_{\alpha}(\mathcal{K};q)$ here are the
reduced invariants which are normalised such that $J_{\alpha}(U;q)$ is unity
as mentioned in footnote 3. The invariants appearing in detcherry2018turaev
follow a different normalisation where $J_{\alpha}(U;q)$ is equal to the
$q$-number $[\alpha+1]$. The Turaev-Viro invariant written in (42) takes care
of this.

$\displaystyle\text{TV}_{k+2}(S^{3}\backslash\mathcal{K}_{m}\,;e^{\frac{\pi
i}{k+2}})$
$\displaystyle=\frac{2}{k+2}\,\sum_{\alpha=0}^{k}\sin^{2}\left(\frac{\pi\alpha+\pi}{k+2}\right)\absolutevalue{J_{\alpha}(\mathcal{K}_{m}\,;e^{\frac{2\pi
i}{k+2}})}^{2}$
$\displaystyle=\frac{2}{k+2}\,\sum_{\alpha=0}^{k}\sin^{2}\left(\frac{\pi\alpha+\pi}{k+2}\right)\absolutevalue{J_{\alpha}(\mathcal{K}\,;e^{\frac{2\pi
i}{k+2}})}^{2m}~{}.$ (42)

We also have the following inequality which is valid for $0\leq\alpha\leq k$ :

$\frac{1}{(k+2)^{2}}\leq\sin^{2}\left(\frac{\pi\alpha+\pi}{k+2}\right)\leq
1~{}.$ (43)

Using (41) and (42), and applying (43), we obtain the following inequalities:

$\frac{2}{(k+2)^{3}}\,\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]\leq\text{TV}_{k+2}(S^{3}\backslash\mathcal{K}_{m}\,;e^{\frac{\pi
i}{k+2}})\leq\frac{2}{k+2}\,\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]~{}.$
(44)

Further, as mentioned in detcherry2018turaev , the growth of
$\text{TV}_{r}(M\,;e^{\frac{\pi i}{r}})$ is expected to be a polynomial in $r$
for 3-manifolds $M$. Thus the above inequality tells that
$\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]$ can grow at most polynomially
in $k$. In particular, taking log, dividing by $k$ and letting $k\to\infty$ in
the expression (44), we get:

$\lim_{k\to\infty}\frac{\ln\,\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]}{k}=\lim_{k\to\infty}\frac{\ln\text{TV}_{k+2}(S^{3}\backslash\mathcal{K}_{m}\,;e^{\frac{\pi
i}{k+2}})}{k}=0~{}.$ (45)

Since the trace $\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]$ in (41) can be
viewed as the SU(2) partition function $Z(M_{\mathcal{K}_{m}})$, we have the
following limit:

$\lim_{k\to\infty}\frac{\ln\,Z(M_{\mathcal{K}_{m}})}{k}=\lim_{k\to\infty}\frac{\ln\text{TV}_{k+2}(S^{3}\backslash\mathcal{K}_{m}\,;e^{\frac{\pi
i}{k+2}})}{k}=0~{}.$ (46)

In the following section, we investigate the large $k$ asymptotics for SO(3)
gauge group.

###  3.4 SO(3) gauge group & $m$-moments of density matrix

Here, we will confine to Chern-Simons theory based on SO(3) gauge group where
the level $k$ is a positive even integer. Since the group SO(3) has rank one,
the integrable representations are given by the Dynkin labels $\alpha$ of the
highest weights in the following range:

$\displaystyle 0\leq\alpha\leq k\quad;\quad\alpha\in\mathbb{Z},\,\,k\in
2\mathbb{Z}~{}.$ (47)

Using the group theoretical fact that the representation of SO(3) with Dynkin
label $\alpha$ is isomorphic to the representation of SU(2) with Dynkin label
$2\alpha$, we can write the list of integrable representations of
$\mathfrak{so}(3)_{k}$ in terms of following Young diagrams:

$\left\\{\,\underbrace{\tiny\hbox{}{\vbox{\offinterlineskip\vbox{\hrule
height=0.3pt\hbox{\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt} \hrule
height=0.3pt}\vspace{-\y@linethick} }}}_{2\alpha}\,\,:\,\,0\leq\alpha\leq
k\right\\}~{}.$ (48)

Hence the basis states of $\mathcal{H}_{T^{2}}$ can be accordingly labeled:

$\text{basis}(\mathcal{H}_{T^{2}})=\left\\{\ket{e_{\alpha}}\,\,:\,\,\alpha\equiv\underbrace{\tiny\hbox{}{\vbox{\offinterlineskip\vbox{\hrule
height=0.3pt\hbox{\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt\hbox
to5.46666pt{\hfil}\vrule height=4.37334pt,width=0.3pt,depth=1.0933pt} \hrule
height=0.3pt}\vspace{-\y@linethick} }}}_{2\alpha}\right\\}~{}.$ (49)

Thus, the state (28) associated with the link $\mathcal{K}\\#T_{2,2}$ can be
written as

$\ket{\mathcal{K}\\#T_{2,2}}=\sum_{\alpha=0}^{k}W_{\alpha}(\mathcal{K}\,;e^{\frac{2\pi
i}{k+1}})\,\ket{e_{\alpha},e_{\alpha}}~{},$ (50)

where $W_{\alpha}(\mathcal{K}\,;q)$ are polynomials in $q$ and are the colored
knot invariants for SO(3) group.555The reduced Chern-Simons invariants of
knots for SO($N$) gauge group are called the colored Kauffman polynomials:
$\text{Kauff}_{\alpha}(\mathcal{K};a,q)$, where the two variables are
$a=q^{N-1}$ and $q=e^{\frac{2\pi i}{k+N-2}}$. For the group SO(3), we define
$W_{\alpha}(\mathcal{K}\,;q)\equiv\text{Kauff}_{\alpha}(\mathcal{K};q^{2},q)$
with $q=e^{\frac{2\pi i}{k+1}}$ which enter as probability amplitudes in (50).
Using the isomorphism between the SO(3) and SU(2) homologies, we can convert
these SO(3) invariants into the colored Jones invariants. The two invariants
are related as (we refer to Nawata:2013mzx  for some explicit examples):

$W_{\alpha}(\mathcal{K}\,;q)=J_{2\alpha}(\mathcal{K}\,;q^{2})~{},$ (51)

where $J_{2\alpha}$ is the colored Jones invariant evaluated for the spin
$\alpha$ representation of SU(2) group. Thus we can rewrite the state in terms
of the colored Jones invariants as:

$\ket{\mathcal{K}\\#T_{2,2}}=\sum_{\alpha=0}^{k}J_{2\alpha}(\mathcal{K}\,;e^{\frac{4\pi
i}{k+1}})\,\ket{e_{\alpha},e_{\alpha}}~{}.$ (52)

The $m$-moment or the trace of the $m^{\text{th}}$ power of the unnormalised
reduced density matrix associated with $\ket{\mathcal{K}\\#T_{2,2}}$ will be:

$\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]=\sum_{\alpha=0}^{k}\absolutevalue{J_{2\alpha}(\mathcal{K}\,;e^{\frac{4\pi
i}{k+1}})}^{2m}~{}.$ (53)

One of the key questions we plan to investigate is the large $k$ behaviour of
these $m$\- moments. We believe that they must capture geometrical features of
the three-manifolds. This will be the focus in the following subsection.

####  3.4.1 Large $k$ asymptotics: The exponential growth

We numerically checked that the colored Jones
$J_{2\alpha}(\mathcal{K}\,;e^{\frac{4\pi i}{k+1}})$ has a maximum value when
the color is $2\alpha=k$ (note that $k$ is a positive even integer). Even
though this observation was validated for some knots, we do not have an
analytic argument.

For the SO(3) Chern-Simons theory where coupling $k\in 2\mathbb{Z}$ and
$\sigma(\mathcal{K}\\#T_{2,2})$ is the unnormalised reduced density matrix
associated with the state $\ket{\mathcal{K}\\#T_{2,2}}$ ($\mathcal{K}$ is a
prime knot and $T_{2,2}$ is the Hopf link), we put forth the following
conjecture.

Conjecture 1. _The large $k$ asymptotics of the trace of the $m^{\text{th}}$
power of the matrix $\sigma(\mathcal{K}\\#T_{2,2})$ captures the hyperbolic
volume of $S^{3}\backslash\mathcal{K}$:_

$\lim_{k\to\infty}\frac{\ln\,\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]}{k}=m\frac{\text{Vol}(S^{3}\backslash\mathcal{K})}{2\pi}~{}.$
(54)

_Proof._ Using our numerical result that the colored Jones
$J_{k}(\mathcal{K}\,;e^{\frac{4\pi i}{k+1}})$ has maximum value, we can write
the following inequality for (53):

$\absolutevalue{J_{k}(\mathcal{K}\,;e^{\frac{4\pi
i}{k+1}})}^{2m}\leq\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]\leq(k+1)\absolutevalue{J_{k}(\mathcal{K}\,;e^{\frac{4\pi
i}{k+1}})}^{2m}~{}.$ (55)

Taking log, dividing by $k$ and then taking $k\to\infty$ limit will give

$\lim_{k\to\infty}\frac{\ln\,\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]}{k}=\lim_{k\to\infty}\frac{\ln\,\absolutevalue{J_{k}(\mathcal{K}\,;e^{\frac{4\pi
i}{k+1}})}^{2m}}{k}~{}.$ (56)

To write the RHS of the above equation as volume, we will use the following
relation between the colored Jones invariant and the Kashaev’s invariant
murakami2001colored :

$\absolutevalue{\text{Kas}_{n}(\mathcal{K}\,;q)}=\absolutevalue{J_{n-1}(\mathcal{K}\,;q)}\quad,\quad
q=\exp(\frac{2\pi i\,a}{n})\quad,\quad\text{gcd}(a,n)=1~{},$ (57)

where $\text{Kas}_{n}(\mathcal{K}\,;q)$ is the Kashaev’s invariant of the knot
$\mathcal{K}$ and the variable $q$ is taken to be a primitive $n^{\text{th}}$
root of unity. In the large $n$ limit, the above Kashaev’s invariant grows
exponentially obeying the following conjecture kashaev1997hyperbolic :

$\lim_{n\to\infty}\frac{\ln\,\absolutevalue{\text{Kas}_{n}(\mathcal{K}\,;e^{\frac{2\pi
ia}{n}})}}{n}=\frac{\text{Vol}(S^{3}\backslash\mathcal{K})}{2\pi a}~{}.$ (58)

That is, its growth rate is determined by the hyperbolic volume of
$S^{3}\backslash\mathcal{K}$. This conjecture has been verified for $a=1$ case
in Ref.kashaev1997hyperbolic ; hikami2003volume  for many hyperbolic knots.
For the SO(3) invariants we are studying, we have $n=(k+1)$, which is odd.
Hence $a=2$ is also allowed in (57) and (58). We have checked the $a=2$ case
of (58) in appendix A for some torus and non-torus knots. From the above
discussion, we can now state that the $m$-moment of the matrix $\sigma$ for
the class of two-component links $\mathcal{K}\\#T_{2,2}$ must obey the
following in the large $k$ limit:

$\lim_{k\to\infty}\frac{\ln\,\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]}{k}=\lim_{k\to\infty}\frac{\ln\,\absolutevalue{J_{k}(\mathcal{K}\,;e^{\frac{4\pi
i}{k+1}})}^{2m}}{k}=2m\frac{\text{Vol}(S^{3}\backslash\mathcal{K})}{4\pi}~{},$
(59)

which gives our conjectured result (54).

Further, following our earlier arguments that
$\text{Tr}[\sigma(\mathcal{K}\\#T_{2,2})]$ is the partition function
$Z(M_{\mathcal{K}})$, the conjecture (54) leads to the following corollary:

Corollary 1. _For any prime knot $\mathcal{K}$, there exists a closed three-
manifold $M_{\mathcal{K}}$ such that the large $k$ asymptotics of the
corresponding SO(3) Chern-Simons partition function $Z(M_{\mathcal{K}})$
determines the hyperbolic volume of $S^{3}\backslash\mathcal{K}$:_

$\lim_{k\to\infty}\frac{\ln\,Z(M_{\mathcal{K}})}{k}=\frac{\text{Vol}(S^{3}\backslash\mathcal{K})}{2\pi}~{}.$
(60)

Thus, even though the explicit topology of $M_{\mathcal{K}}$ is not
visualisable, the leading large $k$ behaviour of the SO(3) Chern-Simons
partition function $Z(M_{\mathcal{K}})$ captures the hyperbolic volume of
$S^{3}\backslash\mathcal{K}$.

Moreover, as we have shown earlier in (37) that the Rényi entropies can be
written in terms of the partition functions $Z(M_{\mathcal{K}_{m}})$, it will
be important to study the large $k$ asymptotics of the Rényi entropies. This
will be the content of the following subsection.

####  3.4.2 Large $k$ limits of Rényi entropies

We have already discussed that the large $k$ asymptotics of
$\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]$ or $Z(M_{\mathcal{K}_{m}})$ is
governed by the asymptotics of
$\absolutevalue{J_{k}(\mathcal{K}\,;e^{\frac{4\pi i}{k+1}})}$. Since
$\absolutevalue{J_{k}(\mathcal{K}\,;e^{\frac{4\pi
i}{k+1}})}=\absolutevalue{\text{Kas}_{k+1}(\mathcal{K}\,;e^{\frac{4\pi
i}{k+1}})}$ from (57), it suffices to obtain the full asymptotics of
$\absolutevalue{\text{Kas}_{k+1}(\mathcal{K}\,;e^{\frac{4\pi i}{k+1}})}$.
Setting $n=(k+1)$, the following functional form for the Kashaev’s invariant
in the large $n$ limit was conjectured in hikami2003volume :

$\frac{2\pi}{n}\ln\absolutevalue{\text{Kas}_{n}(\mathcal{K}\,;q=e^{\frac{2\pi
i}{n}})}\sim c_{1}(\mathcal{K})+c_{2}(\mathcal{K})\frac{2\pi}{n}\ln
n+\frac{c_{3}(\mathcal{K})}{n}+\frac{c_{4}(\mathcal{K})}{n^{2}}~{},$ (61)

where $c_{i}(\mathcal{K})$ are knot dependent constants such that
hikami2003volume :

$c_{1}(\mathcal{K})=\text{Vol}(S^{3}\backslash\mathcal{K})\quad;\quad
c_{2}(\mathcal{K})=\frac{3}{2}~{}.$ (62)

Doing the numerical analysis for some of the non-trivial prime knots, we find
that we can modify (61) for $q=e^{\frac{4\pi i}{n}}$ as following:

$\frac{4\pi}{n}\ln\absolutevalue{\text{Kas}_{n}(\mathcal{K}\,;q=e^{\frac{4\pi
i}{n}})}\sim a_{1}(\mathcal{K})+a_{2}(\mathcal{K})\frac{4\pi}{n}\ln
n+\frac{a_{3}(\mathcal{K})}{n}+\frac{a_{4}(\mathcal{K})}{n^{2}}~{},$ (63)

where $a_{i}(\mathcal{K})$ are constants such that:

$a_{1}(\mathcal{K})=\text{Vol}(S^{3}\backslash\mathcal{K})\quad;\quad
a_{2}(\mathcal{K})=\frac{3}{2}~{}.$ (64)

As a result, we will have the following asymptotics:

$\ln\absolutevalue{J_{k}(\mathcal{K},\,q=e^{\frac{4\pi
i}{k+1}})}\sim{\frac{\text{Vol}(S^{3}\backslash\mathcal{K})}{4\pi}k}+\ln
k^{3/2}+\frac{a_{3}(\mathcal{K})}{4\pi}+\frac{a_{4}(\mathcal{K})}{4\pi k}~{}.$
(65)

From (30) and (36), we will have:

$\frac{Z(M_{\mathcal{K}_{m}})}{\absolutevalue{J_{k}(\mathcal{K},\,e^{\frac{4\pi
i}{k+1}})}^{2m}}=\sum_{\alpha=0}^{k}\absolutevalue{\frac{J_{2\alpha}(\mathcal{K}\,;e^{\frac{4\pi
i}{k+1}})}{J_{k}(\mathcal{K}\,;e^{\frac{4\pi i}{k+1}})}}^{2m}\equiv
X_{m}(\mathcal{K};k)~{}.$ (66)

Using (65), we have following asymptotics for large $k$:

$Z(M_{\mathcal{K}_{m}})\sim
k^{3m}\exp\left[\frac{m\,\text{Vol}(S^{3}\backslash\mathcal{K})}{2\pi}k\right]\times\left(\exp\left[m\frac{a_{3}(\mathcal{K})}{2\pi}+m\frac{a_{4}(\mathcal{K})}{2\pi
k}\right]X_{m}(\mathcal{K};k)\right)~{}.$ (67)

From the numerical computations for several knots, we observe that the term in
the parentheses is convergent as $k\to\infty$. So we define a knot dependent
constant $C_{m}$ as:

$C_{m}(\mathcal{K})=\lim_{k\to\infty}\exp\left[m\frac{a_{3}(\mathcal{K})}{2\pi}+m\frac{a_{4}(\mathcal{K})}{2\pi
k}\right]X_{m}(\mathcal{K};k)~{}.$ (68)

Unfortunately, we do not have an analytical way to compute
$C_{m}(\mathcal{K})$ and we restrict to the numerical computations. To compute
$C_{m}(\mathcal{K})$, we define a function

$F_{m}(\mathcal{K})\equiv
Z(M_{\mathcal{K}_{m}})\exp\left[-\frac{m\,\text{Vol}(S^{3}\backslash\mathcal{K})}{2\pi}k\right]~{}.$
(69)

From the above discussion, we expect the function $F_{m}(\mathcal{K})$ (for a
prime knot $\mathcal{K}$) to grow as $k^{3m}$ for large values of $k$:

$F_{m}(\mathcal{K})\sim C_{m}(\mathcal{K})\,k^{3m}~{}.$ (70)

We first evaluate the values of $F_{m}(\mathcal{K})$ using (69) for
sufficiently large values of $k$ and then use (70) to obtain the constants
$C_{m}(\mathcal{K})$ numerically using the least-square fitting method in
Mathematica. In section 4, we present examples of prime knots up to six
crossings, where $C_{m}(\mathcal{K})$ has been tabulated up to certain decimal
places. With all the numerical results in hand, we propose the following:

Proposition. _The large $k$ asymptotics of SO(3) Chern-Simons partition
function of $M_{\mathcal{K}_{m}}$ is given as following:_

$\displaystyle
Z(M_{\mathcal{K}_{m}})=\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]\sim
C_{m}(\mathcal{K})\,k^{3m}\exp\left(\frac{m\,\text{Vol}(S^{3}\backslash\mathcal{K})}{2\pi}k\right)~{},$
(71)

_where $C_{m}(\mathcal{K})$ is a knot-dependent constant. As a result, the
Rényi entropies converge to a finite value in the limit of $k\to\infty$ and
its value is determined by the constants $C_{m}(\mathcal{K})$:_

$\displaystyle\lim_{k\to\infty}\mathcal{R}_{m}=\frac{1}{1-m}\ln\left[\frac{C_{m}(\mathcal{K})}{C_{1}(\mathcal{K})^{m}}\right]~{}.$
(72)

In the next section, we will present the numerical results validating our
conjecture and proposal. We will restrict to prime knots up to six crossings.

##  4 Numerical results for SO(3) group

In this section, we will present the numerical results for all prime knots up
to six crossings. A list of these knots, along with their hyperbolic volumes,
are given in table 1. As discussed in the previous section, we obtain the
constants $C_{m}(\mathcal{K})$ numerically for various values of $m$. We show
the predicted variation of the function $C_{m}(\mathcal{K})\,k^{3m}$ and
contrast it with the actual computed values of $F_{m}(\mathcal{K})$ from (69).
We further list the proposed $k\to\infty$ limiting values of Rényi entropies
obtained using $C_{m}(\mathcal{K})$ via (72). We justify these predicted
limiting values by plotting the variation of Rényi entropies (calculated from
(37)) with $k$ and showing that the plots tend to converge to the proposed
values.

$3_{1}$ | $4_{1}$ | $5_{1}$ | $5_{2}$ | $6_{1}$ | $6_{2}$ | $6_{3}$
---|---|---|---|---|---|---
$\begin{array}[]{c}\includegraphics[width=43.36464pt]{Knot31.png}\end{array}$ | $\begin{array}[]{c}\includegraphics[width=43.36464pt]{Knot41.png}\end{array}$ | $\begin{array}[]{c}\includegraphics[width=43.36464pt]{Knot51.png}\end{array}$ | $\begin{array}[]{c}\includegraphics[width=43.36464pt]{Knot52.png}\end{array}$ | $\begin{array}[]{c}\includegraphics[width=43.36464pt]{Knot61.png}\end{array}$ | $\begin{array}[]{c}\includegraphics[width=43.36464pt]{Knot62.png}\end{array}$ | $\begin{array}[]{c}\includegraphics[width=43.36464pt]{Knot63.png}\end{array}$
0 | 2.02988 | 0 | 2.82812 | 3.16396 | 4.40083 | 5.69302
Table 1: List of all non-trivial prime knots up to six crossings. The first
row of the table gives the notation of the knots according to the Rolfsen
table rolfsen1990knots . The second row describes the knot diagram and the
third row gives the hyperbolic volume of the knot complement.

For ease of presentation, a list of plots and tables for various knots are
referred in the following:

$\mathcal{K}$ | $C_{m}(\mathcal{K})$ |  $C_{m}(\mathcal{K})$ vs $m$ |  $F_{m}(\mathcal{K})$ vs $k$ | $\lim_{k\to\infty}\mathcal{R}_{m}$ |  $\mathcal{R}_{m}$ vs $k$
---|---|---|---|---|---
$3_{1}$ | Table 2 | Figure 4 | Figure 5 | Table 3 | Figure 6
$4_{1}$ | Table 6 | Figure 10 | Figure 11 | Table 7 | Figure 12
$5_{1}$ | Table 4 | Figure 7 | Figure 8 | Table 5 | Figure 9
$5_{2}$ | Table 8 | Figure 13 | Figure 14 | Table 9 | Figure 15
$6_{1}$ | Table 10 | Figure 16 | Figure 17 | Table 11 | Figure 18
$6_{2}$ | Table 12 | Figure 19 | Figure 20 | Table 13 | Figure 21
$6_{3}$ | Table 14 | Figure 22 | Figure 23 | Table 15 | Figure 24

We will take up the simplest torus knots in the following section to verify
the results of previous section.

###  4.1 Torus knots

####  4.1.1 Trefoil knot: $3_{1}$

This is a torus knot with vanishing hyperbolic volume:
$\text{Vol}(S^{3}\backslash 3_{1})=0$. The proposed form for the function
$F_{m}$

$\displaystyle F_{m}(3_{1})=C_{m}(3_{1})\,k^{3m}$ (73)

is computed using (69) for some sufficiently large values of $k$. In fact, the
least-square method in Mathematica enables us to determine the constant
$C_{m}(3_{1})$ for various values of $m$. Some of these values are listed in
table 2 and the variation of $\ln C_{m}(3_{1})$ with $m$ is shown in figure 4.

$m$ | $C_{m}(3_{1})$
---|---
1 | $1.2508395500\pm 0.0000141876$
2 | $1.2724830787\pm 0.0000133285$
3 | $1.4267959549\pm 0.0000139704$
4 | $1.6058331673\pm 0.0000151173$
5 | $1.8076384699\pm 0.0000167464$
$\vdots$ | $\vdots$
96 | $86270.3224528739\pm 0.3000329135$
97 | $97112.8266267165\pm 0.3310750028$
98 | $109318.0234964881\pm 0.3652932337$
99 | $123057.1771676096\pm 0.4030095261$
100 | $138523.0763324435\pm 0.4445781686$
Table 2: Values of $C_{m}(3_{1})$

Figure 4: Variation of $\ln C_{m}(3_{1})$ with $m$

In figure 5, we plot the function $C_{m}(3_{1})\,k^{3m}$ and compare it with
the numerical values of $F_{m}(3_{1})$ obtained explicitly from (69).

Figure 5: The variation of the function $F_{m}$ with $k$ for $3_{1}$ knot. The
solid line denotes the function $C_{m}\,k^{3m}$ with values of $C_{m}$ given
in table 2. The $\bullet$ indicates the values of $F_{m}$ obtained from
explicit computations using (69).

The predicted $k\to\infty$ values of the Rényi entropies associated with the
state $\ket{3_{1}\\#T_{2,2}}$ can be computed using (72) and we have listed
some of these values in table 3. We also show the variation of the Rényi
entropies with $k$ in the plots in figure 6. We see that these plots tend to
converge to the predicted $k\to\infty$ values of the Rényi entropies. With
this warm-up on trefoil, we present the numerical results for the five
crossing torus knot.

$m$ | $\lim_{k\to\infty}\mathcal{R}_{m}$
---|---
2 | $0.2066597601\pm 0.0000249864$
3 | $0.1580067791\pm 0.0000177041$
4 | $0.1405390447\pm 0.0000154454$
5 | $0.1317633872\pm 0.0000143660$
$\vdots$ | $\vdots$
96 | $0.1065367978\pm 0.0000114619$
97 | $0.1065252390\pm 0.0000114607$
98 | $0.1065139185\pm 0.0000114595$
99 | $0.1065028290\pm 0.0000114583$
100 | $0.1064919634\pm 0.0000114571$
Table 3: The $k\to\infty$ values of the Rényi entropies associated with the
state $\ket{3_{1}\\#T_{2,2}}$.

Figure 6: Variation of Rényi entropies for the state $\ket{3_{1}\\#T_{2,2}}$
with $k$.

####  4.1.2 Solomon’s Seal knot: $5_{1}$

$5_{1}$ is also a torus knot with vanishing hyperbolic volume:
$\text{Vol}(S^{3}\backslash 5_{1})=0$. Similar to our numerical methodology
done for trefoil, we compute $F_{m}(5_{1})$ (69) and using the proposed large
$k$ form

$\displaystyle F_{m}(5_{1})=C_{m}(5_{1})\,k^{3m}~{},$ (74)

and least-square method in Mathematica, we obtained the constant
$C_{m}(5_{1})$ for various values of $m$. Some of these values are listed in
table 4 and the variation of $C_{m}(5_{1})$ with $m$ is shown in figure 7.

$m$ | $C_{m}(5_{1})$
---|---
1 | $1.0962198097\pm 0.0000123719$
2 | $1.0141454043\pm 0.0000110043$
3 | $1.0172196551\pm 0.0000107028$
4 | $1.0228630176\pm 0.0000105694$
5 | $1.0286308590\pm 0.0000104842$
$\vdots$ | $\vdots$
96 | $1.7173551256\pm 0.0000040805$
97 | $1.7270567284\pm 0.0000040224$
98 | $1.7368131320\pm 0.0000039648$
99 | $1.7466246464\pm 0.0000039076$
100 | $1.7564915828\pm 0.0000038509$
Table 4: Values of $C_{m}(5_{1})$

Figure 7: Variation of $C_{m}(5_{1})$ with $m$

In figure 8, we plot the function $C_{m}(5_{1})\,k^{3m}$ and compare it with
the numerical values of $F_{m}(5_{1})$ obtained explicitly from (69).

Figure 8: The variation of the function $F_{m}$ with $k$ for $5_{1}$ knot. The
solid line denotes the function $C_{m}\,k^{3m}$ with values of $C_{m}$ given
in table 4. The $\bullet$ indicates the numerical values of $F_{m}$ obtained
from explicit computations using (69).

The predicted $k\to\infty$ values of the Rényi entropies associated with the
state $\ket{5_{1}\\#T_{2,2}}$ can be computed using (72) and we have listed
some of these values in table 5. We also show the variation of the Rényi
entropies with $k$ in the plots in figure 9. We see that these plots tend to
converge to the predicted $k\to\infty$ values of the Rényi entropies. We have
also done similar numerical validation for other knots with non-trivial
hyperbolic volume which we present in the following subsection.

$m$ | $\lim_{k\to\infty}\mathcal{R}_{m}$
---|---
2 | $0.1696891578\pm 0.0000250446$
3 | $0.1292650485\pm 0.0000177275$
4 | $0.1149551079\pm 0.0000154371$
5 | $0.1077774922\pm 0.0000143357$
$\vdots$ | $\vdots$
96 | $0.0871422756\pm 0.0000114048$
97 | $0.0871328193\pm 0.0000114036$
98 | $0.0871235580\pm 0.0000114023$
99 | $0.0871144857\pm 0.0000114012$
100 | $0.0871055967\pm 0.0000114000$
Table 5: The $k\to\infty$ values of the Rényi entropies associated with the
state $\ket{5_{1}\\#T_{2,2}}$.

Figure 9: Variation of Rényi entropies for the state $\ket{5_{1}\\#T_{2,2}}$
with $k$.

###  4.2 Non-torus knots

####  4.2.1 Figure-eight knot: $4_{1}$

The lowest crossing non-torus knot is the figure-eight knot $4_{1}$ with the
hyperbolic volume $\text{Vol}(S^{3}\backslash 4_{1})=2.029883212819307$. Using
$F_{m}(4_{1})$ (69) and the following proposed large $k$ form,

$\displaystyle F_{m}(4_{1})=C_{m}(4_{1})\,k^{3m}~{},$ (75)

the constant $C_{m}(4_{1})$ was deduced using the least-square method for
various values of $m$. Some of these values are listed in table 6 and the
variation of $\ln C_{m}(4_{1})$ with $m$ is shown in figure 10.

$m$ | $C_{m}(4_{1})$
---|---
1 | $2.5957570692\pm 0.0001620425$
2 | $6.2495177996\pm 0.0004659030$
3 | $15.6076018833\pm 0.0011272905$
4 | $38.9986179208\pm 0.0027607882$
5 | $97.4465517130\pm 0.0067868268$
6 | $243.4918590302\pm 0.0167002962$
7 | $608.4194021638\pm 0.0410857092$
8 | $1520.2754646304\pm 0.1009983598$
9 | $3798.7623179296\pm 0.2480046124$
10 | $9492.1051075026\pm 0.6082069793$
Table 6: Values of $C_{m}(4_{1})$

Figure 10: Variation of $\ln C_{m}(4_{1})$ with $m$

In figure 11, we plot the function $C_{m}(4_{1})\,k^{3m}$ and compare it with
the numerical values of $F_{m}(4_{1})$ obtained explicitly from (69).

Figure 11: The variation of the function $F_{m}$ with $k$ for $4_{1}$ knot.
The solid line denotes the function $C_{m}k^{3m}$ with values of $C_{m}$ given
in table 6. The $\bullet$ indicates the numerical values of $F_{m}$ obtained
from explicit computations.

The predicted $k\to\infty$ values of the Rényi entropies associated with the
state $\ket{4_{1}\\#T_{2,2}}$ can be computed using (72) and we have listed
some of these values in table 7. We also show the variation of the Rényi
entropies with $k$ in the plots in figure 12. We see that these plots tend to
converge to the predicted $k\to\infty$ values of the Rényi entropies. We
present similar analysis and our numerical results for the five crossing non-
torus knot in the following section to validate our conjecture and the
proposal.

$m$ | $\lim_{k\to\infty}\mathcal{R}_{m}$
---|---
2 | $0.0752521224\pm 0.0001454157$
3 | $0.0569382755\pm 0.0001003615$
4 | $0.0506622182\pm 0.0000865149$
5 | $0.0475217594\pm 0.0000799514$
6 | $0.0456371569\pm 0.0000761567$
7 | $0.0443805098\pm 0.0000736947$
8 | $0.0434827000\pm 0.0000719724$
9 | $0.0428091692\pm 0.0000707017$
10 | $0.0422851628\pm 0.0000697266$
Table 7: The $k\to\infty$ values of the Rényi entropies associated with the
state $\ket{4_{1}\\#T_{2,2}}$.

Figure 12: Variation of Rényi entropies for the state $\ket{4_{1}\\#T_{2,2}}$
with $k$.

####  4.2.2 Three-twist knot: $5_{2}$

$F_{m}(5_{2})$ for this five crossing non-torus knot, whose knot complement
volume is $\text{Vol}(S^{3}\backslash 5_{2})=2.828122088330783$, is computed
using (69). Then, from the proposed large $k$ form

$\displaystyle F_{m}(5_{2})=C_{m}(5_{2})\,k^{3m}~{},$ (76)

the constant $C_{m}(5_{2})$ is determined using the least-square method for
various values of $m$. Some of these values are listed in table 8 and the
variation of $\ln C_{m}(5_{2})$ with $m$ is shown in figure 13.

$m$ | $C_{m}(5_{2})$
---|---
1 | $4.5156047311\pm 0.0080192291$
2 | $19.522561281\pm 0.0323263046$
3 | $85.809660629\pm 0.1375626362$
4 | $377.22537837\pm 0.5933078608$
5 | $1658.35734226\pm 2.5705101499$
6 | $7290.66518025\pm 11.154379776$
7 | $32052.9326226\pm 48.419257460$
8 | $140922.2991482\pm 210.12155723$
9 | $619587.6221496\pm 911.29217860$
10 | $2724184.491914\pm 3949.0717321$
Table 8: Values of $C_{m}(5_{2})$

Figure 13: Variation of $\ln C_{m}(5_{2})$ with $m$

In figure 14, we plot the function $C_{m}(5_{2})\,k^{3m}$ and compare it with
the numerical values of $F_{m}(5_{2})$ obtained explicitly from (69).

Figure 14: The variation of the function $F_{m}$ with $k$ for $5_{2}$ knot.
The solid line denotes the function $C_{m}k^{3m}$ with values of $C_{m}$ given
in table 8. The $\bullet$ indicates the numerical values of $F_{m}$ obtained
from explicit computations.

The predicted $k\to\infty$ values of the Rényi entropies associated with the
state $\ket{5_{2}\\#T_{2,2}}$ can be computed using (72) and we have listed
some of these values in table 9. We also show the variation of the Rényi
entropies with $k$ in the plots in figure 15. We see that these plots tend to
converge to the predicted $k\to\infty$ values of the Rényi entropies. In the
rest of this section, we address six crossing non-torus knots to verify our
conjecture and proposal.

$m$ | $\lim_{k\to\infty}\mathcal{R}_{m}$
---|---
2 | $0.0435074467\pm 0.0039188003$
3 | $0.0352428767\pm 0.0027818216$
4 | $0.0324378784\pm 0.0024252028$
5 | $0.0310281855\pm 0.0022534345$
6 | $0.0301769261\pm 0.0021529270$
7 | $0.0296049755\pm 0.0020871156$
8 | $0.0291927116\pm 0.0020407385$
9 | $0.0288803294\pm 0.0020063206$
10 | $0.0286346099\pm 0.0019797771$
Table 9: The $k\to\infty$ values of the Rényi entropies associated with the
state $\ket{5_{2}\\#T_{2,2}}$.

Figure 15: Variation of Rényi entropies for the state $\ket{5_{2}\\#T_{2,2}}$
with $k$.

####  4.2.3 Stevedore knot: $6_{1}$

This is a non-torus knot whose knot complement volume is
$\text{Vol}(S^{3}\backslash 6_{1})=3.16396322888$. In fact, computing
$F_{m}(6_{1})$ (69) and relating to the form

$\displaystyle
F_{m}(6_{1})\equiv\frac{\text{Tr}[\sigma^{m}(6_{1}\\#T_{2,2})]}{\exp\left(\frac{m\,\text{Vol}(S^{3}\backslash
6_{1})}{2\pi}k\right)}=C_{m}(6_{1})\,k^{3m}~{},$ (77)

and using the least-square method, we computed $C_{m}(6_{1})$ for various
values of $m$. Some of these values are listed in table 10 and the variation
of $\ln C_{m}(6_{1})$ with $m$ is shown in figure 16.

$m$ | $C_{m}(6_{1})$
---|---
1 | $4.7923232325\pm 0.0030931604$
2 | $22.3923919843\pm 0.0135258863$
3 | $105.7984752566\pm 0.0616708792$
4 | $499.9052116194\pm 0.2842710891$
5 | $2362.131219566\pm 1.3135290735$
6 | $11161.64299789\pm 6.0672497245$
7 | $52742.37636985\pm 27.982406758$
8 | $249228.8880632\pm 128.78990106$
9 | $1177724.7827384\pm 591.37837422$
10 | $5565389.9109037\pm 2708.8286274$
Table 10: Values of $C_{m}(6_{1})$

Figure 16: Variation of $\ln C_{m}(6_{1})$ with $m$

In figure 17, we plot the function $C_{m}(6_{1})\,k^{3m}$ and compare it with
the numerical values of $F_{m}(6_{1})$ obtained explicitly from (69).

Figure 17: The variation of the function $F_{m}$ with $k$ for $6_{1}$ knot.
The solid line denotes the function $C_{m}k^{3m}$ with values of $C_{m}$ given
in table 10. The $\bullet$ indicates the numerical values of $F_{m}$ obtained
from explicit computations.

The predicted $k\to\infty$ values of the Rényi entropies associated with the
state $\ket{6_{1}\\#T_{2,2}}$ can be computed using (72) and we have listed
some of these values in table 11. We also show the variation of the Rényi
entropies with $k$ in the plots in figure 18. We see that these plots tend to
converge to the predicted $k\to\infty$ values of the Rényi entropies.

$m$ | $\lim_{k\to\infty}\mathcal{R}_{m}$
---|---
2 | $0.0253093645\pm 0.0014252152$
3 | $0.0197549127\pm 0.0010110794$
4 | $0.0178809135\pm 0.0008812152$
5 | $0.0169392514\pm 0.0008186904$
6 | $0.0163706838\pm 0.0007821215$
7 | $0.0159887767\pm 0.0007581881$
8 | $0.0157136437\pm 0.0007413313$
9 | $0.0155053518\pm 0.0007288286$
10 | $0.0153417228\pm 0.0007191926$
Table 11: The $k\to\infty$ values of the Rényi entropies associated with the
state $\ket{6_{1}\\#T_{2,2}}$.

Figure 18: Variation of Rényi entropies for the state $\ket{6_{1}\\#T_{2,2}}$
with $k$.

####  4.2.4 $6_{2}$ knot

The hyperbolic volume is $\text{Vol}(S^{3}\backslash 6_{2})=4.400832516$ for
the complement of the non-torus knot $6_{2}$. We computed $F_{m}(6_{2})$ (69)
and compared with the proposed form

$\displaystyle
F_{m}(6_{2})\equiv\frac{\text{Tr}[\sigma^{m}(6_{2}\\#T_{2,2})]}{\exp\left(\frac{m\,\text{Vol}(S^{3}\backslash
6_{2})}{2\pi}k\right)}=C_{m}(6_{2})\,k^{3m}$ (78)

to deduce $C_{m}(6_{2})$ for various values of $m$ using least-square method.
Some of these values are listed in table 12 and the variation of $\ln
C_{m}(6_{2})$ with $m$ is shown in figure 19.

$m$ | $C_{m}(6_{2})$
---|---
1 | $7.2078662291\pm 0.0114864102$
2 | $50.9252866533\pm 0.0745214114$
3 | $362.4041343538\pm 0.5060376066$
4 | $2579.2833524324\pm 3.4585013119$
5 | $18358.617056\pm 23.58889$
6 | $130681.2420\pm 160.1351$
7 | $930284.933\pm 1080.959$
8 | $6622851.572\pm 7253.540$
9 | $47151678.319\pm 48386.251$
10 | $335713848.188\pm 320930.180$
Table 12: Values of $C_{m}(6_{2})$

Figure 19: Variation of $\ln C_{m}(6_{2})$ with $m$

In figure 20, we plot the function $C_{m}(6_{2})\,k^{3m}$ and compare it with
the numerical values of $F_{m}(6_{2})$ obtained explicitly from (69).

Figure 20: The variation of the function $F_{m}$ with $k$ for $6_{2}$ knot.
The solid line denotes the function $C_{m}k^{3m}$ with values of $C_{m}$ given
in table 12. The $\bullet$ indicates the numerical values of $F_{m}$ obtained
from explicit computations.

The predicted $k\to\infty$ values of the Rényi entropies associated with the
state $\ket{6_{2}\\#T_{2,2}}$ can be computed using (72) and we have listed
some of these values in table 13. We also show the variation of the Rényi
entropies with $k$ in the plots in figure 21. We see that these plots tend to
converge to the predicted $k\to\infty$ values of the Rényi entropies.

$m$ | $\lim_{k\to\infty}\mathcal{R}_{m}$
---|---
2 | $0.0199863319\pm 0.0035070715$
3 | $0.0163794510\pm 0.0024902620$
4 | $0.0151416590\pm 0.0021712926$
5 | $0.0145026174\pm 0.0020177258$
6 | $0.0141042798\pm 0.0019279527$
7 | $0.0138274220\pm 0.0018692517$
8 | $0.0136210151\pm 0.0018279583$
9 | $0.0134595614\pm 0.0017973760$
10 | $0.0133288780\pm 0.0017738427$
Table 13: The $k\to\infty$ values of the Rényi entropies associated with the
state $\ket{6_{2}\\#T_{2,2}}$.

Figure 21: Variation of Rényi entropies for the state $\ket{6_{2}\\#T_{2,2}}$
with $k$.

####  4.2.5 $6_{3}$ knot

Finally, we address similar analysis for $6_{3}$ knot whose knot complement
volume is $\text{Vol}(S^{3}\backslash 6_{3})=5.69302109$. Thus, working out
$F_{m}(6_{3})$ (69) and comparing with

$\displaystyle
F_{m}(6_{3})\equiv\frac{\text{Tr}[\sigma^{m}(6_{3}\\#T_{2,2})]}{\exp\left(\frac{m\,\text{Vol}(S^{3}\backslash
6_{3})}{2\pi}k\right)}=C_{m}(6_{3})\,k^{3m}~{},$ (79)

the constant $C_{m}(6_{3})$ can be deduced for various values of $m$ using the
least-square method. Some of these values are listed in table 14 and the
variation of $\ln C_{m}(6_{3})$ with $m$ is shown in figure 22.

$m$ | $C_{m}(6_{3})$
---|---
1 | $9.5970632893\pm 0.0185978565$
2 | $90.554839418\pm 0.160095501$
3 | $858.76989699\pm 1.44999102$
4 | $8144.95594531\pm 13.23524518$
5 | $77257.4532157\pm 120.6783625$
6 | $732873.448104\pm 1096.079889$
7 | $6952667.48358\pm 9906.38029$
8 | $65963622.8502\pm 89061.0909$
9 | $625870830.613\pm 796416.850$
10 | $5938663602.54\pm 7084757.98$
Table 14: Values of $C_{m}(6_{3})$

Figure 22: Variation of $\ln C_{m}(6_{3})$ with $m$

In figure 23, we plot the function $C_{m}(6_{3})\,k^{3m}$ and compare it with
the numerical values of $F_{m}(6_{3})$ obtained explicitly from (69).

Figure 23: The variation of the function $F_{m}$ with $k$ for $6_{3}$ knot.
The solid line denotes the function $C_{m}k^{3m}$ with values of $C_{m}$ given
in table 14. The $\bullet$ indicates the numerical values of $F_{m}$ obtained
from explicit computations.

The predicted $k\to\infty$ values of the Rényi entropies associated with the
state $\ket{6_{3}\\#T_{2,2}}$ can be computed using (72) and we have listed
some of these values in table 15. We also show the variation of the Rényi
entropies with $k$ in the plots in figure 24. We see that these plots tend to
converge to the predicted $k\to\infty$ values of the Rényi entropies.

These numerical analysis and plots for all the prime knots up to six crossings
indeed validate our conjecture and proposal within SO(3) Chern-Simons theory.

$m$ | $\lim_{k\to\infty}\mathcal{R}_{m}$
---|---
2 | $0.0169586610\pm 0.0042599254$
3 | $0.0144352100\pm 0.0030269172$
4 | $0.0135581550\pm 0.0026399898$
5 | $0.0130967624\pm 0.0024536120$
6 | $0.0128029098\pm 0.0023446021$
7 | $0.0125940093\pm 0.0022732852$
8 | $0.0124347393\pm 0.0022230911$
9 | $0.0123074665\pm 0.0021858981$
10 | $0.0122023867\pm 0.0021572646$
Table 15: The $k\to\infty$ values of the Rényi entropies associated with the
state $\ket{6_{3}\\#T_{2,2}}$.

Figure 24: Variation of Rényi entropies for the state $\ket{6_{3}\\#T_{2,2}}$
with $k$.

##  5 Conclusion

In this paper, our primary goal was to capture the geometrical features of the
three-manifolds from the study of entanglement properties of bi-partite
states. Particularly, we focused on the Chern-Simons theory to study the
topological entanglement between two torus boundaries of a three-manifold
$S^{3}\backslash\mathcal{L}$ (the link complement of a two-component link
$\mathcal{L}$). Interestingly, for a class of two-component links viewed as a
connected sum of a prime knot $\mathcal{K}$ and the Hopf link
($\mathcal{L}=\mathcal{K}\,\\#\,T_{2,2}$), the probability amplitudes of the
state $\ket{\mathcal{K}\,\\#\,T_{2,2}}$ are proportional to the quantum
invariants of the knot $\mathcal{K}$. Thus the trace of the reduced density
matrix (3) is almost similar to the Turaev-Viro invariant zbMATH00167790 ;
detcherry2018turaev  of $S^{3}\backslash\mathcal{K}$. This motivated us to
study the reduced density matrices for this class of two-component links for
both SU(2) and SO(3) gauge groups.

The partial tracing of one of the torus boundary gives the reduced density
matrix $\sigma(\mathcal{K}\\#T_{2,2})$ (29), which provides the starting point
for computing various entanglement measures. The Rényi entropy with index $m$
can be obtained from the $m$-moment or the trace of the $m^{\text{th}}$ power
of the matrix $\sigma(\mathcal{K}\\#T_{2,2})$ using (19). Incidentally, the
trace operation on the reduced density matrix can be viewed as gluing the two
oppositely oriented torus boundaries resulting in a closed three-manifold
$M_{\mathcal{K}}$ giving the Chern-Simons partition function
$Z(M_{\mathcal{K}})=\text{Tr}[\sigma(\mathcal{K}\\#T_{2,2})]$. The subscript
on the three-manifold $M_{\mathcal{K}}$ keeps track of the knot involved in
the two-component link $\mathcal{L}=\mathcal{K}\\#T_{2,2}$. We would like to
mention that the authors of the Ref.Gang:2017cwq  study the perturbative
expansion of the SL$(2,\mathbb{C})$ Chern-Simons partition function of closed
three-manifolds which are obtained from the Dehn filling of link complements.
It would be worthwhile to explore the results of Gang:2017cwq  in the set-up
of our paper. In particular, it will be interesting to see if the manifold
$M_{\mathcal{K}}$ can be viewed as the Dehn filling of certain link complement
which can enable us to obtain the subleading terms in the expansion of
$Z(M_{\mathcal{K}})$. We leave this discussion for future work.

We were able to reproduce similar geometrical results for the higher moments
of the reduced density matrix. Our crucial observation that the replica trick
in the entanglement context is quantified by the connected sums of knots led
to (35) and (36) respectively. Thus we have shown that all the $m$-moments of
the matrix $\sigma(\mathcal{K}\\#T_{2,2})$ are the Chern-Simons partition
functions $Z(M_{\mathcal{K}_{m}})$ where
$\mathcal{K}_{m}=\mathcal{K}\\#\mathcal{K}\ldots\\#\mathcal{K}$ denotes the
connected sum of $m$-copies of $\mathcal{K}$. As a result, the Rényi entropies
with index $m$ can be written in terms of the three-manifold invariants
$Z(M_{\mathcal{K}_{m}})$ as given in (37).

Even though we cannot visualise the explicit topology of $M_{\mathcal{K}}$, we
were able to address the large $k$ asymptotics of $Z(M_{\mathcal{K}})$ for
SU(2) and SO(3) gauge groups. Using the asymptotics of Turaev-Viro invariant
of $S^{3}\backslash\mathcal{K}$ evaluated at $e^{\frac{\pi i}{k+2}}$, we were
able to show in (45), that
$\text{Tr}[\sigma(\mathcal{K}\\#T_{2,2})]=Z(M_{\mathcal{K}})$ for SU(2) group
can grow at most polynomially in $k$. In contrast, the traces show an
exponential growth in $k$ for the SO(3) group. Incorporating the well known
Kashaev’s conjecture kashaev1997hyperbolic , we conjecture that the leading
term of $\ln\text{Tr}[\sigma(\mathcal{K}\\#T_{2,2})]$ and hence $\ln
Z(M_{\mathcal{K}})$ for SO(3) gauge group, captures the hyperbolic volume of
the knot complement $S^{3}\backslash\mathcal{K}$ in the large $k$ limit. These
are given as conjecture (54) and corollary (60) respectively. We have
performed numerical computations for torus and non-torus knots up to six
crossings to validate our conjecture and proposal.

We further propose an asymptotic form for
$\text{Tr}[\sigma^{m}(\mathcal{K}\\#T_{2,2})]=Z(M_{\mathcal{K}_{m}})$ for
SO(3) gauge group, in the large $k$ limit (71). Using such a form, we see that
the Rényi entropies converge to a finite number as $k\to\infty$ (72). We would
like to mention that currently we do not have an understanding of the
geometric or topological interpretation of the large $k$ limiting values of
Rényi entropies, and it will be worthwhile to explore this aspect in the
context of SO(3) Chern-Simons theories.

###### Acknowledgements.

SD is supported by the NSFC grants No. 12050410249 and No. 11975158. PR would
like to thank SERB (MATRICS) MTR/2019/000956 funding. VKS would like to thank
Rama Mishra for helpful discussion and acknowledge the hospitality of
department of Mathematics, IISER, Pune (India) where this work was done during
his visit as visiting fellow. BPM acknowledges the Research Grant for Faculty
under IoE Scheme (number 6031).

##  Appendix A SO(3) Chern-Simons theory and hyperbolic volumes of
$S^{3}\backslash\mathcal{K}$

In the appendix, we will verify the following conjecture for some of the prime
knots:

$\lim_{k\to\infty}\frac{\ln\,\absolutevalue{J_{k}(\mathcal{K}\,;e^{\frac{4\pi
i}{k+1}})}}{k}=\frac{\text{Vol}(S^{3}\backslash\mathcal{K})}{4\pi}~{}.$ (80)

For this, we use the equality between the colored Jones and Kashaev’s
invariant. In particular, setting $n=(k+1)$ with $k\in 2\mathbb{Z}$, we have:

$\absolutevalue{\text{Kas}_{n}(\mathcal{K}\,;q=e^{\frac{4\pi
i}{n}})}=\absolutevalue{J_{n-1}(\mathcal{K}\,;q=e^{\frac{4\pi i}{n}})}~{}.$
(81)

This has already been discussed in section 3 around eq.(57). So, we study the
asymptotics of the above Kashaev’s invariant for prime knots up to six
crossings using the stationary phase approximation prescribed in
kashaev1997hyperbolic ; hikami2003volume . This approach works for those knots
whose invariant can be written in terms of $q$-Pochhammer symbols

$(q)_{\ell}=\prod_{j=1}^{\ell}(1-q^{j})~{}.$ (82)

We consider the limit

$q=\exp(\frac{4\pi i}{n})\quad,\quad n\to\infty~{}.$ (83)

The aim is to obtain the maximum summand that dominates the asymptotic of the
Kashaev’s invariant in the above limit. For this, the summations appearing in
the Kashaev’s invariant are converted into integral expressions of the
following form:

$\text{Kas}_{n}(\mathcal{K}\,;q=e^{\frac{4\pi i}{n}})\sim\iint\ldots\int
dx_{1}dx_{2}\ldots dx_{p}\,\exp(\frac{n}{4\pi
i}\,V_{\mathcal{K}}(x_{1},\ldots,x_{p}))~{},$ (84)

where the integration variable $x_{i}=q^{a_{i}}$ with $a_{i}$ being the
summation variable in the Kashaev’s invariant. The potential $V_{\mathcal{K}}$
is formally obtained by replacing each of the $q$-Pochhammer symbols appearing
in the Kashaev’s invariant by their asymptotic expressions. This is done by
approximating the $q$-Pochhammer by a dilogarithm function as shown below:

$\displaystyle\ln\,(q)_{\ell}$
$\displaystyle=\sum_{j=1}^{\ell}\ln(1-e^{\frac{4\pi ij}{n}})\sim\frac{n}{4\pi
i}\int_{0}^{\frac{4\pi i\ell}{n}}\ln(1-e^{t})\,dt$
$\displaystyle\Longrightarrow(q)_{\ell}$ $\displaystyle\sim\exp[\frac{n}{4\pi
i}\left(\frac{\pi^{2}}{6}-\text{Li}_{2}(q^{\ell})\right)]~{}.$ (85)

Once the potential $V_{\mathcal{K}}(x_{1},\ldots,x_{p})$ is known, we can
apply the stationary phase approximation for the integral (84) in the large
$n$ limit, and obtain a saddle point solution $(x_{1}^{0},\ldots,x_{p}^{0})$
which satisfies:

$\left.\frac{\partial\,V_{\mathcal{K}}(x_{1},\ldots,x_{p})}{\partial
x_{j}}\right|_{(x_{1},\ldots,x_{p})=(x_{1}^{0},\ldots,x_{p}^{0})}=0,\quad\forall\,\,j=1,2,\ldots,p~{}.$
(86)

As a result, we can write the following limit:

$\lim_{n\to\infty}\frac{4\pi}{n}\ln\text{Kas}_{n}(\mathcal{K}\,;e^{\frac{4\pi
i}{n}})=-i\,V_{\mathcal{K}}(x_{1}^{0},\ldots,x_{p}^{0})~{}.$ (87)

The real part of the right hand side or the imaginary part of the potential is
expected to coincide with the hyperbolic volume of
$S^{3}\backslash\mathcal{K}$:

$\real[-i\,V_{\mathcal{K}}(x_{1}^{0},\ldots,x_{p}^{0})]=\imaginary[V_{\mathcal{K}}(x_{1}^{0},\ldots,x_{p}^{0})]=\text{Vol}(S^{3}\backslash\mathcal{K})~{}.$
(88)

Thus, finally we can arrive at

$\lim_{n\to\infty}\frac{\ln\absolutevalue{\text{Kas}_{n}(\mathcal{K}\,;e^{\frac{4\pi
i}{n}})}}{n}=\lim_{n\to\infty}\frac{\ln\absolutevalue{J_{n-1}(\mathcal{K}\,;e^{\frac{4\pi
i}{n}})}}{n}=\frac{\text{Vol}(S^{3}\backslash\mathcal{K})}{4\pi}~{}.$ (89)

In the following, we will verify (88) and (89) for prime knots up to six
crossings.

###  $\bullet$ $S^{3}\backslash 3_{1}$

The colored Jones polynomial associated with $3_{1}$ knot colored by an SU(2)
irrep with Dynkin label $\alpha$ is given as:

$J_{\alpha}(3_{1},\,q)=\sum_{j=-\alpha/2}^{\alpha/2}\left(\frac{q^{-\frac{3\alpha^{2}}{2}-\frac{5\alpha}{2}+6j^{2}-5j+1}-q^{-\frac{3\alpha^{2}}{2}-\frac{5\alpha}{2}+6j^{2}+j}}{q^{\alpha+1}-1}\right)~{}.$
(90)

The Kashaev’s invariant for the $3_{1}$ knot is given as,

$\text{Kas}_{n}(3_{1}\,;q)=\sum_{j=0}^{n-1}(q)_{j}~{}.$ (91)

We have numerically verified that:

$\absolutevalue{J_{n-1}(3_{1},\,e^{\frac{4\pi
i}{n}})}=\absolutevalue{\text{Kas}_{n}(3_{1}\,;e^{\frac{4\pi
i}{n}})}\quad,\quad n=1,3,5,7,\ldots~{}.$ (92)

Using (85) and setting $x=q^{j}$ with $q=e^{\frac{4\pi i}{n}}$, we will have
the following expression in large $n$ limit:

$\text{Kas}_{n}(3_{1}\,;e^{\frac{4\pi i}{n}})\sim\int dx\,\exp(\frac{n}{4\pi
i}\,V_{3_{1}}(x))~{},$ (93)

where the potential is given as

$V_{3_{1}}(x)=\frac{\pi^{2}}{6}-\text{Li}_{2}(x)~{}.$ (94)

The saddle point is $x^{0}=0$ and corresponding potential is given as:

$V_{3_{1}}(x^{0})=\frac{\pi^{2}}{6}\quad\Longrightarrow\quad\imaginary[V_{3_{1}}(x^{0})]=0~{}.$
(95)

This gives the vanishing hyperbolic volume of $S^{3}\backslash 3_{1}$ which is
expected since $3_{1}$ is a torus knot. This verifies (89).

###  $\bullet$ $S^{3}\backslash 4_{1}$

The Jones polynomial associated with $4_{1}$ knot colored by an SU(2) irrep
with Dynkin label $\alpha$ is given as:

$J_{\alpha}(4_{1},\,q)=\sum_{a=0}^{\alpha}\prod_{j=1}^{a}\left(q^{\frac{1}{2}(\alpha-j+1)}-q^{-\frac{1}{2}(\alpha-j+1)}\right)\left(q^{\frac{1}{2}(\alpha+j+1)}-q^{-\frac{1}{2}(\alpha+j+1)}\right)$
(96)

The Kashaev’s invariant for the $4_{1}$ knot is given as kashaev1997hyperbolic
,

$\text{Kas}_{n}(4_{1}\,;q)=\sum_{j=0}^{n-1}\absolutevalue{(q)_{j}}^{2}~{}.$
(97)

We have numerically verified that:

$\absolutevalue{J_{n-1}(4_{1},\,e^{\frac{4\pi
i}{n}})}=\absolutevalue{\text{Kas}_{n}(4_{1}\,;e^{\frac{4\pi
i}{n}})}\quad,\quad n=1,3,5,7,\ldots~{}.$ (98)

Defining $x=q^{j}$ with $q=e^{\frac{4\pi i}{n}}$, and using (85), we will have
the following expression in the large $n$ limit:

$\text{Kas}_{n}(4_{1}\,;e^{\frac{4\pi i}{n}})\sim\int dx\,\exp(\frac{n}{4\pi
i}\,V_{4_{1}}(x))~{},$ (99)

where the potential is given as kashaev1997hyperbolic :

$V_{4_{1}}(x)=\text{Li}_{2}(x^{-1})-\text{Li}_{2}(x)~{}.$ (100)

The saddle point is $x^{0}=e^{-\frac{i\pi}{3}}$ at which we have:

$\imaginary[V_{4_{1}}(x^{0})]\approx 2.0298832128193072500424...~{}.$ (101)

This coincides with the hyperbolic volume of $S^{3}\backslash 4_{1}$ and
verifies (89).

###  $\bullet$ $S^{3}\backslash 5_{1}$

The colored Jones polynomial associated with $5_{1}$ knot colored by an SU(2)
irrep with Dynkin label $\alpha$ is given as:

$J_{\alpha}(5_{1},\,q)=\sum_{j=-\alpha/2}^{\alpha/2}\left(\frac{q^{-\frac{5\alpha^{2}}{2}-\frac{9\alpha}{2}+10j^{2}-7j+1}-q^{-\frac{5\alpha^{2}}{2}-\frac{9\alpha}{2}+10j^{2}+3j}}{q^{\alpha+1}-1}\right)~{}.$
(102)

The Kashaev’s invariant for the $5_{1}$ knot is given as,

$\text{Kas}_{n}(5_{1}\,;q)=\sum_{\begin{subarray}{c}j_{1},\,j_{2}=0\\\
0\,\leq\,j_{1}+j_{2}\,\leq\,n-1\end{subarray}}^{n-1}q^{-j_{1}j_{2}}\,(q)_{j_{1}+j_{2}}~{}.$
(103)

We have numerically verified that:

$\absolutevalue{J_{n-1}(5_{1},\,e^{\frac{4\pi
i}{n}})}=\absolutevalue{\text{Kas}_{n}(5_{1}\,;e^{\frac{4\pi
i}{n}})}\quad,\quad n=1,3,5,7,\ldots~{}.$ (104)

Defining $x_{1}=q^{j_{1}}$ and $x_{2}=q^{j_{2}}$ with $q=e^{\frac{4\pi
i}{n}}$, using (85) and noting that

$q^{-j_{1}j_{2}}=e^{-\frac{4\pi ij_{1}j_{2}}{n}}\sim e^{-\frac{n}{4\pi i}(\ln
x_{1})(\ln x_{2})}~{},$ (105)

we will have the following expression in the large $n$ limit:

$\text{Kas}_{n}(5_{1}\,;e^{\frac{4\pi i}{n}})\sim\iint
dx_{1}dx_{2}\,\exp(\frac{n}{4\pi i}\,V_{5_{1}}(x_{1},x_{2}))~{},$ (106)

where the potential is given as

$V_{5_{1}}(x_{1},x_{2})=\frac{\pi^{2}}{6}-\text{Li}_{2}(x_{1}x_{2})-(\ln
x_{1})(\ln x_{2})~{}.$ (107)

The saddle points are given as:

$(x_{1}^{0},x_{2}^{0})=(-\tfrac{\sqrt{5}+1}{2},-\tfrac{\sqrt{5}+1}{2})\,,\,(\tfrac{\sqrt{5}-1}{2},\tfrac{\sqrt{5}-1}{2})~{}.$
(108)

At both of these saddle points, we find that
$\imaginary[V_{5_{1}}(x_{1}^{0},x_{2}^{0})]=0$. This gives the vanishing
hyperbolic volume of $S^{3}\backslash 5_{1}$ which is expected since $5_{1}$
is a torus knot. This verifies (89).

###  $\bullet$ $S^{3}\backslash 5_{2}$

The Jones polynomial associated with $5_{2}$ knot colored by an SU(2) irrep
with Dynkin label $\alpha$ is given as:

$J_{\alpha}(5_{2},\,q)=\sum_{a_{1}=0}^{\alpha}\sum_{a_{2}=0}^{a_{1}}\frac{(-1)^{a_{2}}\,q^{a_{1}+\frac{1}{2}a_{2}(5a_{2}+3)}(1-q^{2a_{2}+1})\binom{a_{1}}{a_{2}}_{q}(q^{-\alpha};q)_{a_{1}}(q^{\alpha+2};q)_{a_{1}}}{(q^{a_{2}+1};q)_{a_{1}+1}}~{},$
(109)

where the following definitions are used:

$(z;q)_{\ell}=\prod_{j=0}^{\ell-1}(1-zq^{j})\quad,\quad\binom{x}{y}_{q}=\frac{(q;q)_{x}}{(q;q)_{y}\,(q;q)_{x-y}}~{}.$
(110)

The Kashaev’s invariant for the $5_{2}$ knot is given as kashaev1997hyperbolic
,

$\text{Kas}_{n}(5_{2}\,;q)=\sum_{j_{1}=0}^{n-1}\sum_{j_{2}=0}^{j_{1}}\,q^{-(j_{1}+1)j_{2}}\,\frac{(q)_{j_{1}}\,(q)_{j_{1}}}{(q)_{j_{2}}^{*}}~{}.$
(111)

We have numerically checked that

$\absolutevalue{J_{n-1}(5_{2},\,e^{\frac{4\pi
i}{n}})}=\absolutevalue{\text{Kas}_{n}(5_{2}\,;e^{\frac{4\pi
i}{n}})}\quad,\quad n=1,3,5,7,\ldots~{}.$ (112)

Setting $x_{1}=q^{j_{1}}$ and $x_{2}=q^{j_{2}}$ with $q=e^{\frac{4\pi i}{n}}$,
using (85) and (105), we have the following asymptotic for the Kashaev’s
invariant in the large $n$ limit:

$\text{Kas}_{n}(5_{2}\,;e^{\frac{4\pi i}{n}})\sim\iint
dx_{1}dx_{2}\,\exp(\frac{n}{4\pi i}\,V_{5_{2}}(x_{1},x_{2}))~{},$ (113)

where the potential is given as kashaev1997hyperbolic :

$V_{5_{2}}(x_{1},x_{2})=\frac{\pi^{2}}{2}-2\text{Li}_{2}(x_{1})-\text{Li}_{2}({x_{2}}^{-1})-(\ln
x_{1})(\ln x_{2})~{}.$ (114)

The saddle point is approximately given as:

$\left(\begin{array}[]{c}x_{1}^{0}\\\ x_{2}^{0}\\\
\end{array}\right)=\left(\begin{array}[]{c}0.3376410213776269870195456-0.5622795120623012438991821i\\\
0.1225611668766536199752455+0.7448617666197442365931704i\\\
\end{array}\right)$ (115)

at which the imaginary part of the potential is given as:

$\imaginary[V_{5_{2}}(x_{1}^{0},x_{2}^{0})]\approx
2.82812208833078316276390...~{}.$ (116)

This coincides with the hyperbolic volume of $S^{3}\backslash 5_{2}$ and
verifies (89).

###  $\bullet$ $S^{3}\backslash 6_{1}$

The Jones polynomial associated with $6_{1}$ knot colored by an SU(2) irrep
with Dynkin label $\alpha$ is given as:

$J_{\alpha}(6_{1},\,q)=\sum_{a_{1}=0}^{\alpha}\sum_{a_{2}=0}^{a_{1}}\left(q^{-\alpha
a_{1}+a_{2}^{2}+a_{2}-a_{1}}\frac{(q;q)_{a_{1}}(q;q)_{\alpha}\left(q^{\alpha+2};q\right)_{a_{1}}}{(q;q)_{a_{1}-a_{2}}(q;q)_{a_{2}}(q;q)_{\alpha-
a_{1}}}\right)~{}.$ (117)

The Kashaev’s invariant for the $6_{1}$ knot is given as kashaev1997hyperbolic
,

$\text{Kas}_{n}(6_{1}\,;q)=\sum_{\begin{subarray}{c}j_{1},\,j_{2},\,j_{3}=0\\\
j_{1}+j_{2}\,\leq\,j_{3}\end{subarray}}^{n-1}\,q^{(j_{3}-j_{1}-j_{2})(j_{3}-j_{1}+1)}\frac{\absolutevalue{(q)_{j_{3}}}^{2}}{(q)_{j_{1}}\,(q)_{j_{2}}^{*}}\,~{}.$
(118)

We have numerically checked that

$\absolutevalue{J_{n-1}(6_{1},\,e^{\frac{4\pi
i}{n}})}=\absolutevalue{\text{Kas}_{n}(6_{1}\,;e^{\frac{4\pi
i}{n}})}\quad,\quad n=1,3,5,7,\ldots~{}.$ (119)

Setting $x_{1}=q^{j_{1}}$, $x_{2}=q^{j_{2}}$, $x_{3}=q^{j_{3}}$ and using
(85), we have the following asymptotic for the Kashaev’s invariant in the
large $n$ limit:

$\text{Kas}_{n}(6_{1}\,;e^{\frac{4\pi i}{n}})\sim\iiint
dx_{1}dx_{2}dx_{3}\,\exp(\frac{n}{4\pi i}\,V_{6_{1}}(x_{1},x_{2},x_{3}))~{},$
(120)

where the potential is given as kashaev1997hyperbolic :

$\displaystyle
V_{6_{1}}(x_{1},x_{2},x_{3})=\text{Li}_{2}(x_{1})-\text{Li}_{2}(x_{2}^{-1})+\text{Li}_{2}(x_{3}^{-1})-\text{Li}_{2}(x_{3})+\ln(\frac{x_{1}}{x_{3}})\ln(\frac{x_{1}x_{2}}{x_{3}})-2\pi
i\ln(\frac{x_{1}}{x_{3}})~{}.$ (121)

The saddle point is approximately given as:

$\left(\begin{array}[]{c}x_{1}^{0}\\\ x_{2}^{0}\\\
x_{3}^{0}\end{array}\right)=\left(\begin{array}[]{c}0.17384979383679558571428+1.06907189987572611850590i\\\
0.32204184201246211276846+0.15777973787085281762619i\\\
0.27872641157714581676289-0.48341992018615350391557i\end{array}\right)$ (122)

at which the imaginary part of the potential is given as:

$\imaginary[V_{6_{1}}(x_{1}^{0},x_{2}^{0})]\approx
3.163963228883143983991...~{}.$ (123)

This coincides with the hyperbolic volume of $S^{3}\backslash 6_{1}$ and
verifies (89).

###  $\bullet$ $S^{3}\backslash 6_{2}$

The Jones polynomial associated with $6_{2}$ knot colored by an SU(2) irrep
with Dynkin label $\alpha$ is given as:

$\displaystyle J_{\alpha}(6_{2},\,q)$
$\displaystyle=\sum_{a_{1}=0}^{\alpha}\sum_{a_{2}=0}^{a_{1}}\sum_{a_{3}=0}^{a_{2}}\left((-1)^{a_{2}+a_{3}}\,q^{\frac{1}{2}(-2\alpha
a_{1}-a_{2}^{2}+2a_{1}a_{2}+a_{2}+a_{3}^{2}+a_{3})}\binom{a_{1}}{a_{2}}_{q}\,\binom{a_{2}}{a_{3}}_{q}\,(q;q)_{a_{2}}\right.$
$\displaystyle\left.(q^{a_{3}+1};q)_{a_{1}-a_{2}}\,\binom{\alpha}{a_{1}}_{q}\,(q^{\alpha+2};q)_{a_{1}}\right)~{}.$
(124)

The Kashaev’s invariant for the $6_{2}$ knot is given as hikami2003volume ,

$\text{Kas}_{n}(6_{2}\,;q)=\sum_{\begin{subarray}{c}j_{1},\,j_{2},\,j_{3}=0\\\
j_{1}\,\leq\,j_{2}\\\
0\,\leq\,j_{1}+j_{3}\,\leq\,n-1\end{subarray}}^{n-1}\,q^{-j_{1}(j_{2}+j_{3}+1)}\,\frac{(q)_{j_{2}}\,(q)_{j_{2}}\,(q)_{j_{1}+j_{3}}}{\absolutevalue{(q)_{j_{1}}}^{2}\,(q)_{j_{2}-j_{1}}}~{}.$
(125)

We have numerically checked that

$\absolutevalue{J_{n-1}(6_{2},\,e^{\frac{4\pi
i}{n}})}=\absolutevalue{\text{Kas}_{n}(6_{2}\,;e^{\frac{4\pi
i}{n}})}\quad,\quad n=1,3,5,7,\ldots~{}.$ (126)

Setting $x_{1}=q^{j_{1}}$, $x_{2}=q^{j_{2}}$, $x_{3}=q^{j_{3}}$ and using
(85), we have the following asymptotic for the Kashaev’s invariant in the
large $n$ limit:

$\text{Kas}_{n}(6_{2}\,;e^{\frac{4\pi i}{n}})\sim\iiint
dx_{1}dx_{2}dx_{3}\,\exp(\frac{n}{4\pi i}\,V_{6_{2}}(x_{1},x_{2},x_{3}))~{},$
(127)

where the potential is given as hikami2003volume :

$\displaystyle
V_{6_{2}}(x_{1},x_{2},x_{3})=\text{Li}_{2}(x_{1})-\text{Li}_{2}(x_{1}^{-1})+\text{Li}_{2}(x_{1}^{-1}x_{2})-2\text{Li}_{2}(x_{2})-\text{Li}_{2}(x_{1}x_{3})-(\ln
x_{1})(\ln x_{2}x_{3})+\frac{\pi^{2}}{3}~{}.$ (128)

The saddle point is approximately given as:

$\left(\begin{array}[]{c}x_{1}^{0}\\\ x_{2}^{0}\\\
x_{3}^{0}\end{array}\right)=\left(\begin{array}[]{c}0.0904326688828523150727+1.6028830694139448442811i\\\
-0.23270516544550685831671-1.09381177109577462363025i\\\
-0.96491338528654243537970-0.62189628352233029773000i\end{array}\right)$ (129)

at which the imaginary part of the potential is given as:

$\imaginary[V_{6_{2}}(x_{1}^{0},x_{2}^{0})]\approx
4.400832516123046101441...~{}.$ (130)

This coincides with the hyperbolic volume of $S^{3}\backslash 6_{2}$ and
verifies (89).

###  $\bullet$ $S^{3}\backslash 6_{3}$

The Jones polynomial associated with $6_{3}$ knot colored by an SU(2) irrep
with Dynkin label $\alpha$ is given as:

$\displaystyle J_{\alpha}(6_{3},\,q)$
$\displaystyle=\sum_{a_{1}=0}^{\alpha}\sum_{a_{2}=0}^{a_{1}}\sum_{a_{3}=0}^{a_{2}}\left((-1)^{a_{1}+a_{3}}\,q^{\frac{1}{2}\left(-a_{1}\left(2\alpha+2a_{2}+1\right)+a_{1}^{2}+a_{3}^{2}-2a_{2}+a_{3}\right)}\binom{a_{1}}{a_{2}}_{q}\,\binom{a_{2}}{a_{3}}_{q}\,(q;q)_{a_{2}}\right.$
$\displaystyle\left.(q^{a_{3}+1};q)_{a_{1}-a_{2}}\,\binom{\alpha}{a_{1}}_{q}\,(q^{\alpha+2};q)_{a_{1}}\right)~{}.$
(131)

The Kashaev’s invariant for the $6_{3}$ knot is given as hikami2003volume ,

$\text{Kas}_{n}(6_{3}\,;q)=\sum_{\begin{subarray}{c}j_{1},\,j_{2},\,j_{3}=0\\\
0\,\leq\,j_{1}+j_{2}+j_{3}\,\leq\,n-1\end{subarray}}^{n-1}\,q^{(j_{1}+1)(j_{2}-j_{3})}\,\frac{\absolutevalue{(q)_{j_{1}+j_{2}+j_{3}}}^{2}\,(q)_{j_{1}+j_{2}}^{*}\,(q)_{j_{1}+j_{3}}}{\absolutevalue{(q)_{j_{2}}}^{2}\,\absolutevalue{(q)_{j_{3}}}^{2}}~{}.$
(132)

We have numerically checked that

$\absolutevalue{J_{n-1}(6_{3},\,e^{\frac{4\pi
i}{n}})}=\absolutevalue{\text{Kas}_{n}(6_{3}\,;e^{\frac{4\pi
i}{n}})}\quad,\quad n=1,3,5,7,\ldots~{}.$ (133)

Setting $x_{1}=q^{j_{1}}$, $x_{2}=q^{j_{2}}$, $x_{3}=q^{j_{3}}$ and using
(85), we have the following asymptotic for the Kashaev’s invariant in the
large $n$ limit:

$\text{Kas}_{n}(6_{3}\,;e^{\frac{4\pi i}{n}})\sim\iiint
dx_{1}dx_{2}dx_{3}\,\exp(\frac{n}{4\pi i}\,V_{6_{3}}(x_{1},x_{2},x_{3}))~{},$
(134)

where the potential is given as hikami2003volume :

$\displaystyle V_{6_{3}}(x_{1},x_{2},x_{3})$
$\displaystyle=\text{Li}_{2}(x_{2})-\text{Li}_{2}(x_{2}^{-1})+\text{Li}_{2}(x_{3})-\text{Li}_{2}(x_{3}^{-1})+\text{Li}_{2}(x_{1}^{-1}x_{2}^{-1})-\text{Li}_{2}(x_{1}x_{3})$
$\displaystyle-\text{Li}_{2}(x_{1}x_{2}x_{3})+\text{Li}_{2}(x_{1}^{-1}x_{2}^{-1}x_{3}^{-1})+(\ln
x_{1})\ln(x_{2}/x_{3})~{}.$ (135)

The saddle point is approximately given as:

$\left(\begin{array}[]{c}x_{1}^{0}\\\ x_{2}^{0}\\\
x_{3}^{0}\end{array}\right)=\left(\begin{array}[]{c}0.20432287067507999865586-0.97890355220475859397659i\\\
1.6083782859756256503564+0.5587518814119368864517i\\\
0.55478836677297306027131+0.19273391491468578161798i\end{array}\right)$ (136)

at which the imaginary part of the potential is given as:

$\imaginary[V_{6_{3}}(x_{1}^{0},x_{2}^{0})]\approx
5.693021091281300765112...~{}.$ (137)

This coincides with the hyperbolic volume of $S^{3}\backslash 6_{3}$ and
verifies (89).

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