Explicit formulae for Chern-Simons invariants of the hyperbolic $J(2n,-2m)$ knot orbifolds

TL;DR

This paper derives explicit formulas for the Chern-Simons invariants of hyperbolic double twist knot orbifolds, enhancing understanding of their geometric and topological properties.

Contribution

It provides explicit formulae for Chern-Simons invariants of hyperbolic double twist knot orbifolds using the Schl"afli formula, a novel computational approach.

Findings

Explicit formulas for Chern-Simons invariants derived

Application of Schl"afli formula to cone-manifold structures

Enhanced understanding of hyperbolic knot orbifolds

Abstract

We calculate the Chern-Simons invariants of the hyperbolic double twist knot orbifolds using the Schl\"{a}fli formula for the generalized Chern-Simons function on the family of cone-manifold structures of double twist knots.

Figures

Tables

Table 1. Table 6.1. Chern-Simons invariant of X 2 ​ n 2 ​ m superscript subscript 𝑋 2 𝑛 2 𝑚 X_{2n}^{2m} for n 𝑛 n between 1 1 1 and 4 4 4 , m 𝑚 m between 1 1 1 and 4 4 4 with n ≥ m 𝑛 𝑚 n\geq m except amphicheiral knots.

2n	2m	${\alpha }_0$	$\text{cs}\left(X_{2n}^{2m}\right)$
2	2	2.094395102393195	0
4	2	2.574140778131840	0.34402298
6	2	2.750685152010280	0.27786688
8	2	2.843209532683532	0.24222232
4	4	2.847642272262783	0
6	4	2.942465754372979	0.42782933
8	4	2.990939179603150	0.38923730
6	6	3.007517657179940	0
8	6	3.040474611156828	0.46103929
8	8	3.065453796328835	0

Table 2. Table 6.2. Chern-Simons invariant of the hyperbolic orbifold, cs ​ ( X 2 ​ n 2 ​ m ​ ( 2 ​ π k ) ) ​ ( mod ​ 1 k ​ if  k  is even or  mod ​ 1 2 ​ k ​ if  k  is odd ) cs superscript subscript 𝑋 2 𝑛 2 𝑚 2 𝜋 𝑘 mod 1 𝑘 if  k  is even or  mod 1 2 𝑘 if  k  is odd \text{{cs}}\left(X_{2n}^{2m}(\frac{2\pi}{k})\right)\left(\text{mod}\ \frac{1}{k}\ \text{if $k$ is even or }\text{mod}\ \frac{1}{2k}\ \text{if $k$ is odd}\right) for n 𝑛 n between 1 1 1 and 4 4 4 , m 𝑚 m between 1 1 1 and 4 4 4 with n ≥ m 𝑛 𝑚 n\geq m , and for k 𝑘 k between 3 3 3 and 10 10 10 , and of its cyclic covering, cs ​ ( M k ​ ( X 2 ​ n 2 ​ m ) ) cs subscript 𝑀 𝑘 superscript subscript 𝑋 2 𝑛 2 𝑚 \text{{cs}}\left(M_{k}(X_{2n}^{2m})\right) except amphicheiral knots.

$k$ $\text{cs}\left(X_4^2\left(\frac{2\pi }{k}\right)\right)$ $\text{cs}\left(M_k\left(X_4^2\right)\right)$ 3 0.0875301 0.26259 4 0.144925 0.579699 5 0.0784576 0.392288 6 0.0351571 0.210943 7 0.00506505 0.0354553 8 0.108039 0.864313 9 0.0218112 0.196301 10 0.0530574 0.530574

$k$ $\text{cs}\left(X_6^2\left(\frac{2\pi }{k}\right)\right)$ $\text{cs}\left(M_k\left(X_6^2\right)\right)$ 3 0.0449535 0.13486 4 0.0876043 0.350417 5 0.0165337 0.0826684 6 0.138167 0.829004 7 0.0120078 0.0840545 8 0.0430876 0.3447 9 0.012125 0.109125 10 0.0876213 0.876213

Equations

$$\text{cs}\left(X_{2n}^{2m}\left(\frac{2\pi }{k}\right)\right)\equiv \frac{1}{2}\text{cs}\left(L\left(4nm+1,2n(2m-1)+1\right)\right)$$

$$+\frac{1}{4{\pi }^2}{\int }_{\frac{2\pi }{k}}^{{\alpha }_0}Im\left(2\ast \log \left(-\frac{M^2(S_n(v)-S_{n-1}(v))-(S_{n-1}(v)-S_{n-2}(v))}{(S_n(v)-S_{n-1}(v))-M^2(S_{n-1}(v)-S_{n-2}(v))}\right)\right)d\alpha$$

$$+\frac{1}{4{\pi }^2}{\int }_{{\alpha }_0}^{\pi }Im\left(\log \left(-\frac{M^2(S_n(v_1)-S_{n-1}(v_1))-(S_{n-1}(v_1)-S_{n-2}(v_1))}{(S_n(v_1)-S_{n-1}(v_1))-M^2(S_{n-1}(v_1)-S_{n-2}(v_1))}\right)\right)d\alpha$$

$$+\frac{1}{4{\pi }^2}{\int }_{{\alpha }_0}^{\pi }Im\left(\log \left(-\frac{M^2(S_n(v_2)-S_{n-1}(v_2))-(S_{n-1}(v_2)-S_{n-2}(v_2))}{(S_n(v_2)-S_{n-1}(v_2))-M^2(S_{n-1}(v_2)-S_{n-2}(v_2))}\right)\right)d\alpha ,$$

$${\pi }_1(X_{2n}^{2m})=⟨s,t|s{w}^m{t}^{-1}{w}^{-m}=1⟩,$$

$$S_n(v)=\sum _{0\le i\le \lfloor \frac{n}{2}\rfloor }(-1{)}^i(\genfrac{}{}{0ex}{}{n-i}{i})v^{n-2i}$$

$$V^k=\left[\begin{array}{cc}{S}_k(v)-d{S}_{k-1}(v)& b{S}_{k-1}(v)\\ c{S}_{k-1}(v)& S_k(v)-a{S}_{k-1}(v)\end{array}\right]$$

$$\begin{array}[]{ccccc}\rho(s)=\left[\begin{array}[]{cc}M&1\\ 0&M^{-1}\end{array}\right]\text{,}\ \ \ \rho(t)=\left[\begin{array}[]{cc}M&0\\ 2-v&M^{-1}\end{array}\right],\end{array}$$

$$\begin{array}[]{cc}c=\left[\begin{array}[]{cc}0&-\left(\sqrt{2-v}\right)^{-1}\\ \sqrt{2-v}&0\end{array}\right].\end{array}$$

$${\varphi }_{2n}^{2m}(x,M)=S_m(z)+\left[-1+x{S}_{n-1}(v)\left(S_n(v)+(1-v)S_{n-1}(v)\right)\right]S_{m-1}(z).$$

$$T^{-1}S$$

$$T{S}^{-1}$$

$$(T^{-1}S{)}^n$$

$$(T{S}^{-1}{)}^n$$

$$W=(T^{-1}S{)}^n(T{S}^{-1}{)}^n=\left[\begin{array}{cc}{W}_{11}& W_{12}\\ (2-v)W_{12}& W_{22}\end{array}\right]\text{where}$$

$$W_{11}$$

$$W_{12}$$

$$W_{22}$$

$$z=W_{11}+W_{22}$$

$$+(2M^{-2}+2M^2-2v-M^{-2}v-M^{2}v+v^2)S_{n-1}^2(v)$$

$$=2+(v-2)x{S}_{n-1}^2(v).$$

$$(W{)}^m$$

$$\text{tr}(S{W}^{m}c)/\sqrt{2-v}\left(=(M-M^{-1})W_{12}{S}_{m-1}(z)+S_m(z)-W_{11}{S}_{m-1}(z)\right)$$

$$\tilde{W}=\left[\begin{array}{cc}{\tilde{W}}_{22}& {\tilde{W}}_{12}\\ (2-v){\tilde{W}}_{12}& {\tilde{W}}_{11}\end{array}\right]$$

$${\tilde{W}}_{11}$$

$${\tilde{W}}_{12}$$

$${\tilde{W}}_{22}$$

$$L=-\frac{M^2(S_n(v)-S_{n-1}(v))-(S_{n-1}(v)-S_{n-2}(v))}{(S_n(v)-S_{n-1}(v))-M^2(S_{n-1}(v)-S_{n-2}(v))}.$$

$$I\left(X_{2n}^{2m}(\alpha )\right)=\frac{1}{2}{\int }_{\Gamma (S^3-T_{2n}-s)}Q-\frac{1}{4\pi }\tau (s,{\Gamma }^{\prime })-\frac{1}{4\pi }\left(\frac{\beta \alpha }{2\pi }\right),$$

$$Q=\frac{1}{4{\pi }^2}\left({\theta }_{12}\wedge {\theta }_{13}\wedge {\theta }_{23}+{\theta }_{12}\wedge {\Omega }_{12}+{\theta }_{13}\wedge {\Omega }_{13}+{\theta }_{23}\wedge {\Omega }_{23}\right),$$

$$\tau (s,{\Gamma }^{\prime })=-{\int }_{{\Gamma }^{\prime }(s)}{\theta }_{23},$$

$$dI\left(X_{2n}^{2m}(\alpha )\right)=-\frac{1}{4{\pi }^2}\beta d\alpha .$$

$$(x_1,x_2)=\left(2-2\cos \left(\frac{\pi (2m+1)}{4nm+1}\right),2-2\cos \left(\frac{\pi (2m-1)}{4nm+1}\right)\right)$$

$$\text{cs}\left(X_{2n}^{2m}\left(\frac{2\pi }{k}\right)\right)\equiv I\left(X_{2n}^{2m}\left(\frac{2\pi }{k}\right)\right)\left(\text{mod}\frac{1}{k}\right)$$

$$\equiv I\left(X_{2n}^{2m}(\pi )\right)+\frac{1}{4{\pi }^2}{\int }_{\frac{2\pi }{k}}^{\pi }\beta d\alpha \left(\text{mod}\frac{1}{k}\right)$$

$$\equiv \frac{1}{2}\text{cs}\left(L(4nm+1,2n(2m-1)+1)\right)$$

$$+\frac{1}{4{\pi }^2}{\int }_{\frac{2\pi }{k}}^{{\alpha }_0}Im\left(2\ast \log \left(-\frac{M^2(S_n(v)-S_{n-1}(v))-(S_{n-1}(v)-S_{n-2}(v))}{(S_n(v)-S_{n-1}(v))-M^2(S_{n-1}(v)-S_{n-2}(v))}\right)\right)d\alpha$$

Full text

We calculate the Chern-Simons invariants of the hyperbolic double twist knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of cone-manifold structures of double twist knots.

1. Introduction

Chern-Simons invariants of hyperbolic knot orbifolds are computed explicitly for a few infinite families in [HL, HL2, HLMR] using the ‘‘Schläfli formula’’.

In this paper, we present the explicit formulae for Chern-Simons invariants of the hyperbolic double twist knot orbifolds and we present them numerically for some of double twist knot orbifolds. A brief history of Chern-Simons invariant can be found in [HL, HL2, HLMR]. A double twist knot is denoted by $C(2n,2m)$ according to Conway notation or by $J(2n,-2m)$ according to Hoste-Shanahan notation. Figure 2 presents $C(2n,2m)$ for $m,n>0$.

For a two-bridge hyperbolic link, there exists an angle $\alpha_{0}\in[\frac{2\pi}{3},\pi)$ for each link $K$ such that the cone-manifold $K(\alpha)$ is hyperbolic for $\alpha\in(0,\alpha_{0})$, Euclidean for $\alpha=\alpha_{0}$, and spherical for $\alpha\in(\alpha_{0},\pi]$ [P2, HLM1, K1, PW]. We will use the Chern-Simons invariant of the lens space $L\left(4nm+1,2n(2m-1)+1\right)$ calculated in [HLM2]. The following theorem gives the Chern-Simons invariant formulae for the hyperbolic $C(2n,2m)$ knots. Let $S_{k}(v)$ be the Chebychev polynomials defined by $S_{0}(v)=1$, $S_{1}(v)=v$ and $S_{k}(v)=vS_{k-1}(v)-S_{k-2}(v)$ for all integers $k$.

Theorem 1.1**.**

Let $X_{2n}^{2m}\!(\alpha)$ be the hyperbolic cone-manifold with underlying space $S^{3}$ and a singular set $C(2n,2m)$ of cone-angle $0\leq\alpha<\alpha_{0}$. Let $k$ be a positive integer such that $k$-fold cyclic covering of $X_{2n}^{2m}(\frac{2\pi}{k})$ is hyperbolic. Then the Chern-Simons invariant of $X_{2n}^{2m}(\frac{2\pi}{k})$ (mod $\frac{1}{k}$ if $k$ is even or mod $\frac{1}{2k}$ if $k$ is odd) is given by the following formula:

[TABLE]

where for $M=e^{\frac{i\alpha}{2}}$, $x$ , $x_{1}$, and $x_{2}$ are zeroes of Riley-Mednykh polynomial $\phi_{2n}^{2m}(x,M)$ in Theorem 2.5. As $\alpha$ decreases to $\alpha_{0}$ both $x_{1}$ and $x_{2}$ approach a common value $x$. One of $x_{1}$ and $x_{2}$ comes from the component of $x$, and the other comes from the component of $\bar{x}$. $v$ satisfies $\text{{Im}}((S_{n}(v)-S_{n-1}(v))\overline{(S_{n-1}(v)-S_{n-2}(v))})\geq 0$ [Tran3, Lemma 3.9].

2. $C(2n,2m)$ knots

A general reference for this section is  [HS]. A knot with $2n$ right-handed vertical crossings and $2m$ left-handed horizontal crossings as in Figure 2.1 is $C(2n,2m)$ knot according to Conway’s notation. One can easily check that the slope of $C(2n,2m)$ is $2m/(4nm+1)$ which is equivalent to the knot with slope $\left(2n(2m-1)+1\right)/(4nm+1)$ [S1].

We will use the following presentation of the fundamental group of $C(2n,2m)$ knot(equivalently, $J(2n,-2m)$ knot) in [HS]. In [HS], Hoste and Shanahan asked whether their presentation of the fundamental group for double twist knots can be derived from Schubert’s canonical 2-bridge diagram or not. The following proposition can also be obtained by reading off the fundamental group from the Schubert normal form of $C(2n,2m)$ with slope $2m/(4nm+1)$ [S1, R1] which answers Hoste-Shanahan’s question completely for $C(2n,2m)$ knots. Let$X_{2n}^{2m}$ be $S^{3}\backslash C(2n,2m)$.

Proposition 2.1**.**

[HLMR, Proposition2.2]*  [R1, S1]*

[TABLE]

where $w=(t^{-1}s)^{n}(ts^{-1})^{n}$.

2.2. The Chebychev polynomial

Let $S_{k}(v)$ be the Chebychev polynomials defined by $S_{0}(v)=1$, $S_{1}(v)=v$ and $S_{k}(v)=vS_{k-1}(v)-S_{k-2}(v)$ for all integers $k$. The following explicit formula for $S_{k}(v)$ can be obtained by solving the above recurrence relation [Tran2].

[TABLE]

for $n\geq 0$, $S_{n}(v)=-S_{-n-2}(v)$ for $n\leq-2$, and $S_{-1}(v)=0$. The following proposition 2.3 can be proved using the Cayley-Hamilton theorem [Tran].

Proposition 2.3**.**

[Tran, Proposition 2.4]* Suppose $V=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\in\textnormal{SL}_{2}(\mathbb{C}).$ Then*

[TABLE]

where $v=\textnormal{tr}(V)=a+d$.

2.4. The Riley-Mednykh polynomial

Let

[TABLE]

and let

[TABLE]

Then from the above Proposition 2.3, we get the following Theorem 2.5. Let $\rho(s)=S$, $\rho(t)=T$ and $\rho(w)=W$. Then $\textnormal{tr}(T^{-1}S)=v=\textnormal{tr}(TS^{-1})$. Let $v=x+M^{2}+M^{-2}$. Theorem 2.5 can be found in [Tran3]. We include the proof for readers’ convenience.

Theorem 2.5**.**

[Tran3]* $\rho$ is a representation of $\pi_{1}(X_{2n}^{2m})$ if and only if $x$ is a root of the following Riley-Mednykh polynomial,*

[TABLE]

• Proof.

Since

[TABLE]

[TABLE]

Hence

[TABLE]

[TABLE]

Let $z=\text{Tr}(W)$. Then, since $S_{n}^{2}(v)-vS_{n}(v)S_{n-1}(v)+S_{n-1}^{2}(v)=1$ (by [Tran1, Lemma 2.1] or by induction),

[TABLE]

By Proposition 2.3, we have

[TABLE]

Therefore

[TABLE]

gives $\phi_{2n}^{2m}(x,M)$ [HLMR]. ∎

3. Longitude

Let $l=w^{m}(w^{*})^{m}$, where $w^{*}$ is the word obtained by reversing $w$. Then $l$ is the longitude which is null-homologus in $X_{2n}^{2m}$. Recall $\rho(w)=W$. Let $\widetilde{W}=\rho(w^{*})$. It is easy to see that $\widetilde{W}$ can be written as

[TABLE]

where $\tilde{W}_{ij}$ is obtained by $W_{ij}$ by replacing $M$ with $M^{-1}$. Similar computation was introduced in [HS]. Hence,

[TABLE]

The following lemma was introduced in [HS] with slightly different coordinates.

Let $L=\rho(l)_{11}$(the left upper entry of $\rho(l)$).

Lemma 3.1**.**

[HS]*

$W_{21}L+\tilde{W}_{21}=0$.*

Theorem 3.2**.**

[TABLE]

• Proof.

By directly computing $W_{21}L+\tilde{W}_{21}=0$ in Lemma 3.1 and substituting $S_{n}(v)+S_{n-2}(v)$ for $vS_{n-1}(v)$, the theorem follows. ∎

4. Schläfli formula for the generalized Chern-Simons function

The general references for this section are [HLM3, HLM2, Y1, MeyRub1, HL, HL2] and [HLMR].

In [HLM3], Hilden, Lozano, and Montesinos-Amilibia defined the generalized Chern-Simons function on the oriented cone-manifold structures which matches up with the Chern-Simons invariant when the cone-manifold is the Riemannian manifold.

Below, we briefly introduce the generalized Chern-Simons function on the family of $C(2n,2m)$ cone-manifold structures. For an oriented knot $C(2n,2m)$, we orient its chosen meridian $s$ such that the orientation of $s$ followed by the orientation of $C(2n,2m)$ coincides with orientation of $\mathbb{S}^{3}$. Here, we use the definition of the lens space in [HLM2] so that we can have the right orientation when it is combined with the following frame field. On the Riemannian manifold $\mathbb{S}^{3}-C(2n,2m)-s$ we choose a special frame field $\Gamma=(e_{1},e_{2},e_{3})$ which is an orthonormal frame field such that for each point $x$ near $C(2n,2m)$, $e_{1}(x)$ has the direction given by knot’s orientation, $e_{2}(x)$ has the tangent direction of the meridian curve, and $e_{3}(x)$ has the knot to point $x$ direction. Such a special frame field always exists by Proposition $3.1$ of [HLM3]. From $\Gamma$ we obtain an orthonormal frame field $\Gamma_{\alpha}$ on $X_{2n}^{2m}\!(\alpha)-s$ by the Gram-Schmidt orthogonalization process with respect to the Riemannian structure of the cone manifold $X_{2n}^{2m}\!(\alpha)$. Moreover, it can be made special by deforming it in a neighbourhood of the singular set $C(2n,2m)$ and $s$, if necessary. Thus, $\Gamma^{\prime}$ is an extension of $\Gamma$ to $S^{3}-C(2n,2m)$. To the cone-manifold $X_{2n}^{2m}\!(\alpha)$, we assign a real number

[TABLE]

where $-2\pi\leq\beta\leq 2\pi$ is the angle of the lifted holonomy of the singular locus of $X_{2n}^{2m}\!(\alpha)$, $Q$ is the Chern-Simons form:

[TABLE]

and

[TABLE]

where ($\theta_{ij}$) is the connection $1$-form, ($\Omega_{ij}$) is the curvature $2$-form of the Riemannian connection on $X_{2n}^{2m}\!(\alpha)$ and the integral is over the orthonormalizations of the same frame field. When $\alpha=\frac{2\pi}{k}$ for some positive integer, $I\left(X_{2n}^{2m}\!\!\left(\frac{2\pi}{k}\right)\right)$ (mod $\frac{1}{k}$ if $k$ is even or mod $\frac{1}{2k}$ if $k$ is odd) is independent of the frame field $\Gamma$ and of the representative in the equivalence class $\overline{\beta}$ and hence becomes an invariant of the orbifold $X_{2n}^{2m}\!\!\left(\frac{2\pi}{k}\right)$. The quantity $I\left(X_{2n}^{2m}\!\!\left(\frac{2\pi}{k}\right)\right)$ (mod $\frac{1}{k}$ if $k$ is even or mod $\frac{1}{2k}$ if $k$ is odd) is called the Chern-Simons invariant of the orbifold and is denoted by $\text{{cs}}\left(X_{2n}^{2m}\!\!\left(\frac{2\pi}{k}\right)\right)$.

We have the following ‘‘Schläfli formula’’ for the generalized Chern-Simons function on the family of $C(2n,2m)$ cone-manifold structures.

Theorem 4.1**.**

( [HLM2, Theorem 1.2])  For a family of geometric cone-manifold structures, $X_{2n}^{2m}\!(\alpha)$, and differentiable functions $\alpha(t)$ and $\beta(t)$ of $t$ we have

[TABLE]

5. Proof of the theorem 1.1

For $n\geq 1$ and $M=e^{i\frac{\alpha}{2}}$, $\phi_{2n}^{2m}(x,M)$ have $2mn$ component zeros. The component which passes through

[TABLE]

at $\alpha=\pi$ is the geometric component by  [HLM2, Theorem 2.1]. Note that $2-x_{1}>0$ and $2-x_{2}>0$. For each $C(2n,2m)$, there exists an angle $\alpha_{0}\in[\frac{2\pi}{3},\pi)$ such that $C(2n,2m)$ is hyperbolic for $\alpha\in(0,\alpha_{0})$, Euclidean for $\alpha=\alpha_{0}$, and spherical for $\alpha\in(\alpha_{0},\pi]$ [P2, HLM1, K1, PW]. Denote by $D(X_{2n}^{2m}\!(\alpha))$ be the set of zeros of the discriminant of $\phi_{2n}^{2m}(x,e^{i\frac{\alpha}{2}})$ over $x$. Then $\alpha_{0}$ will be one of $D(X_{2n}^{2m}\!(\alpha))$.

On the geometric component we can calculate the Chern-Simons invariant of an orbifold $X_{2n}^{2m}(\frac{2\pi}{k})$ (mod $\frac{1}{k}$ if $k$ is even or mod $\frac{1}{2k}$ if $k$ is odd), where $k$ is a positive integer such that $k$-fold cyclic covering of $X_{2n}^{2m}(\frac{2\pi}{k})$ is hyperbolic:

[TABLE]

where the second equivalence comes from Theorem 4.1 and the third equivalence comes from the fact that $I\left(X_{2n}^{2m}(\pi)\right)\equiv\frac{1}{2}\text{{cs}}\left(L(4nm+1,2n(2m-1)+1)\right)$ $\left(\text{mod }\frac{1}{2}\right)$, Theorem 3.2, and geometric interpretations of hyperbolic and spherical holonomy representations.

The following theorem gives the Chern-Simons invariant of the Lens space $L(4nm+1,2n(2m-1)+1)$.

Theorem 5.1**.**

( [HLM2, Theorem 1.3])

[TABLE]

6. Chern-Simons invariants of the hyperbolic $C(2n,2m)$ knot orbifolds and of its cyclic coverings

The table 6.1 gives the approximate Chern-Simons invariant of $C(2n,2m)$ for $n$ between $1$ and $4$, $m$ between $1$ and $4$ with $n\geq m$. Since $C(2,2)$, $C(4,4)$, $C(6,6)$, $C(8,8)$ are amphicheiral knots, their Chern-Simons invariants are $zero$ as expected. We used Simpson’s rule for the approximation with $2\times 10^{4}$ ($10^{4}$ in Simpson’s rule) intervals from [math] to $\alpha_{0}$ and $2\times 10^{4}$ ($10^{4}$ in Simpson’s rule) intervals from $\alpha_{0}$ to $\pi$. The table 6.2 gives the approximate Chern-Simons invariant of the hyperbolic orbifold, $\text{{cs}}\left(X_{2n}^{2m}(\frac{2\pi}{k})\right)$ for $n$ between $1$ and $4$, $m$ between $1$ and $4$ with $n\geq m$, and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{{cs}}\left(M_{k}(X_{2n}^{2m})\right)$ except amphicheiral knots. We used Simpson’s rule for the approximation with $2\times 10^{2}$ ($10^{2}$ in Simpson’s rule) intervals from $2\pi/k$ to $\alpha_{0}$ and $2\times 10^{2}$ ($10^{2}$ in Simpson’s rule) intervals from $\alpha_{0}$ to $\pi$.

We used Mathematica for the calculations. We record here that our data in Table 6.1 and those obtained from SnapPy [SnapPy] match up up to existing decimal points and our data in Table 6.2. For computational reasons, we need $\alpha_{0}$. $\alpha_{0}$ is the bifercation point of the geometric solution of the Riley-Mednykh polynomial as described in Theorem 1.1.

Note that the Chern-Simons invariant of the hyperbolic orbifold $\text{{cs}}\left(X_{2n}^{2m}(\frac{2\pi}{k})\right)$ is only defined modulo $\frac{1}{k}$ [HLM2, Theorem 1.4] and we only get modulo $\frac{1}{2k}$ for $k$ odd [HLM2, Theorem 1.4].