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Quantum knot invariants

Research in the Mathematical Sciences20185:11

https://doi.org/10.1007/s40687-018-0127-3

Received: 31 August 2017

Accepted: 13 November 2017

Published: 8 February 2018

Abstract

This is a survey talk on one of the best known quantum knot invariants, the colored Jones polynomial of a knot, and its relation to the algebraic/geometric topology and hyperbolic geometry of the knot complement. We review several aspects of the colored Jones polynomial, emphasizing modularity, stability and effective computations. The talk was given in the Mathematische Arbeitstagung June 24–July 1, 2011.

Keywords

Jones polynomialKnotsQuantum topologyVolume conjectureNahm sumsStabilityModularityModular formsMock-modular forms q-Holonomic sequence q-Series

Mathematics Subject Classification

Primary 57N10Secondary 57M25

1 The Jones polynomial of a knot

Quantum knot invariants are powerful numerical invariants defined by quantum field theory with deep connections to the geometry and topology in dimension three [59]. This is a survey talk on the various limits of the colored Jones polynomial [41], one of the best known quantum knot invariants. This is a 25-year-old subject that contains theorems and conjectures in disconnected areas of mathematics. We chose to present some old and recent conjectures on the subject, emphasizing two recent aspects of the colored Jones polynomial, Modularity and Stability and their illustration by effective computations. Zagier’s influence on this subject is profound, and several results in this talk are joint work with him. Of course, the author is responsible for any mistakes in the presentation. We thank Don Zagier for enlightening conversations, for his hospitality and for his generous sharing of his ideas with us.

The Jones polynomial \(J_L(q) \in {\mathbb {Z}}[q^{\pm 1/2}]\) of an oriented link L in 3-space is uniquely determined by the linear relations [41]
The Jones polynomial has a unique extension to a polynomial invariant \(J_{L,c}(q)\) of links L together with a coloring c of their components that are colored by positive natural numbers that satisfy the following rules
$$\begin{aligned} J_{L \cup K, c \cup \{N+1\}}(q)= & {} J_{L \cup K^{(2)}, c \cup \{N,2\} }(q)- J_{L \cup K, c \cup \{N-1\}}(q), \qquad N \ge 2, \\ J_{L \cup K, c \cup \{1\}}(q)= & {} J_{L, c}(q), \\ J_{L,\{2,\dots ,2\}}(q)= & {} J_L(q), \end{aligned}$$
where \((L \cup K, c \cup \{N\})\) denotes a link with a distinguished component K colored by N and \(K^{(2)}\) denotes the 2-parallel of K with zero framing. Here, a natural number N attached to a component of a link indicates the N-dimensional irreducible representation of the Lie algebra \(\mathfrak {sl}(2,{\mathbb {C}})\). For a detailed discussion on the polynomial invariants of links that come from quantum groups, see [40, 56, 57].

The above relations make clear that the colored Jones polynomial of a knot encodes the Jones polynomials of the knot and its 0-framed parallels.

2 Three limits of the colored Jones polynomial

In this section, we will list three conjectures, the MMR Conjecture (proven), the Slope Conjecture (mostly proven) and the AJ Conjecture (less proven). These conjectures relate the colored Jones polynomial of a knot with the Alexander polynomial, with the set of slopes of incompressible surfaces and with the \(\mathrm {PSL}(2,{\mathbb {C}})\) character variety of the knot complement.

2.1 The colored Jones polynomial and the Alexander polynomial

We begin by discussing a relation of the colored Jones polynomial of a knot with the homology of the universal abelian cover of its complement. The homology \(H_1(M,{\mathbb {Z}})\simeq {\mathbb {Z}}\) of the complement \(M=S^3\setminus K\) of a knot K in 3-space is independent of the knot K. This allows us to consider the universal abelian cover \(\widetilde{M}\) of M with deck transformation group \({\mathbb {Z}}\), and with homology \(H_1(\widetilde{M},{\mathbb {Z}})\) a \({\mathbb {Z}}[t^{\pm 1}]\) module. As it is well known, this module is essentially torsion and its order is given by the Alexander polynomial \(\Delta _K(t) \in {\mathbb {Z}}[t^{\pm 1}]\) of K [53]. The Alexander polynomial does not distinguish knots from their mirrors and satisfies \(\Delta _K(1)=1\).

There are infinitely many pairs of knots (for instance \((10_{22},10_{35})\) in the Rolfsen table [3, 53]) with equal Jones polynomial but different Alexander polynomial. On the other hand, the colored Jones polynomial determines the Alexander polynomial. This so-called Melvin–Morton–Rozansky Conjecture was proven in [4] and states that
$$\begin{aligned} {\hat{J}}_{K,n}(e^{\hbar }) =\sum _{i \ge j \ge 0} a_{K,ij} \hbar ^i n^j \in {\mathbb {Q}}[[n,\hbar ]] \end{aligned}$$
(1)
and
$$\begin{aligned} \sum _{i=0}^\infty a_{K,ij} \hbar ^i=\frac{1}{\Delta _K(e^{\hbar })} \in {\mathbb {Q}}[[\hbar ]]. \end{aligned}$$
Here \({\hat{J}}_{K,n}(q)=J_{K,n}(q)/J_{\text {Unknot},n}(q) \in {\mathbb {Z}}[q^{\pm 1}]\) is a normalized form of the colored Jones polynomial. The above conjecture is a statement about formal power series. A stronger analytic version is known [26, Thm.1.3]; namely, for every knot K there exists an open neighborhood \(U_K\) of \(0 \in {\mathbb {C}}\) such that for all \(\alpha \in U_K\) we have
$$\begin{aligned} \lim _n J_{K,n}(e^{\alpha /n})=\frac{1}{\Delta _K(e^{\alpha })}, \end{aligned}$$
where the left-hand side is a sequence of analytic functions of \(\alpha \in U_K\) converging uniformly on each compact subset of \(U_K\) to the function of the right-hand side. More is known about the summation of the series (1) along a fixed diagonal \(i=j+k\) for fixed k, both on the level of formal power series and on the analytic counterpart. For further details, the reader may consult [26] and references therein.

2.2 The colored Jones polynomial and slopes of incompressible surfaces

In this section, we discuss a conjecture relating the degree of the colored Jones polynomial of a knot K with the set \(\mathrm {bs}_K\) of boundary slopes of incompressible surfaces in the knot complement \(M=S^3\setminus K\). Although there are infinitely many incompressible surfaces in M, it is known that \(\mathrm {bs}_K \subset {\mathbb {Q}}\cup \{1/0\}\) is a finite set [38]. Incompressible surfaces play an important role in geometric topology in dimension three, often accompanied by the theory of normal surfaces [37]. From our point of view, incompressible surfaces are a tropical limit of the colored Jones polynomial, corresponding to an expansion around \(q=0\) [20].

The Jones polynomial of a knot is a Laurent polynomial in one variable q with integer coefficients. Ignoring most information, one can consider the degree \(\delta _K(n)\) of \({\hat{J}}_{K,n+1}(q)\) with respect to q. Since \(({\hat{J}}_{K,n}(q))\) is a q-holonomic sequence [25], it follows that \(\delta _K\) is a quadratic quasi-polynomial [18]. In other words, we have
$$\begin{aligned} \delta _K(n)=c_K(n) n^2 + b_K(n) n + a_K(n), \end{aligned}$$
where \(a_K, b_K, c_K: {\mathbb {N}}\longrightarrow {\mathbb {Q}}\) are periodic functions. In [19], the author formulated the Slope Conjecture.

Conjecture 2.1

For all knots K, we have
$$\begin{aligned} 4 c_K({\mathbb {N}}) \subset \mathrm {bs}_K. \end{aligned}$$
The motivating example for the Slope Conjecture was the case of the \((-2,3,7)\) pretzel knot, where we have [19, Ex.1.4]
$$\begin{aligned} \delta _{(-2,3,7)}(n) = \left[ \frac{37}{8} n^2 + \frac{17}{2} n \right] = \frac{37}{8} n^2 + \frac{17}{2} n + a(n), \end{aligned}$$
where a(n) is a periodic sequence of period 4 given by \(0,-1/8,-1/2,-1/8\) if \(n\equiv 0,1,2,3 \bmod 4\), respectively. In addition, we have
$$\begin{aligned} \mathrm {bs}_{(-2,3,7)}=\{0,16, 37/2, 20\}. \end{aligned}$$
In all known examples, \(c_K({\mathbb {N}})\) consists of a single element, the so-called Jones slope. How the colored Jones polynomial selects some of the finitely many boundary slopes is a challenging and interesting question. The Slope Conjecture is known for all torus knots, all alternating knots and all knots with at most 8 crossings [19] as well as for all adequate knots [15] and all 2-fusion knots [10].

2.3 The colored Jones polynomial and the \(\mathrm {PSL}(2,{\mathbb {C}})\) character variety

In this section, we discuss a conjecture relating the colored Jones polynomial of a knot K with the moduli space of \(\mathrm {SL}(2,{\mathbb {C}})\)-representations of M, restricted to the boundary of M. Ignoring 0-dimensional components, the latter is a one-dimensional plane curve. To formulate the conjecture, we need to recall that the colored Jones polynomial \({\hat{J}}_{K,n}(q)\) is q-holonomic [25], i.e., it satisfies a non-trivial linear recursion relation
$$\begin{aligned} \sum _{j=0}^d a_j(q,q^n) {\hat{J}}_{K,n+j}(q)=0 \end{aligned}$$
(2)
for all n where \(a_j(u,v) \in {\mathbb {Z}}[u^{\pm 1},v^{\pm 1}]\) and \(a_d \ne 0\). q-holonomic sequences were introduced by Zeilberger [66], and a fundamental theorem (multisums of q-proper hypergeometric terms are q-holonomic) was proven in [61] and implemented in [52]. Using two operators M and L which act on a sequence f(n) by
$$\begin{aligned} (M f)(n)=q^n f(n), \qquad (L f)(n)=f(n+1), \end{aligned}$$
we can write the recursion (2) in operator form
$$\begin{aligned} P \cdot {\hat{J}}_{K}=0 \qquad \text {where} \qquad P=\sum _{j=0}^d a_j(q,M) L^j. \end{aligned}$$
It is easy to see that \(LM=qML\) and ML generate the q-Weyl algebra. Although a q-holonomic sequence is annihilated by many operators P (for instance, if it is annihilated by P, then it is also annihilated by any left multiple QP of P), one can choose a canonical recursion, denoted by \(A_K(M,L,q) \in {\mathbb {Z}}[q,M]\langle L \rangle /(LM-qML)\) and called non-commutative A-polynomial of K, which is a knot invariant [16]. The reason for this terminology is the potential relation with the A-polynomial \(A_K(M,L)\) of K [6]. The latter is defined as follows.
Let \(X_M=\mathrm {Hom}(\pi _1(M),\mathrm {SL}(2,{\mathbb {C}}))/{\mathbb {C}}\) denote the moduli space of flat \(\mathrm {SL}(2,{\mathbb {C}})\) connections on M. We have an identification
$$\begin{aligned} X_{\partial M} \simeq ({\mathbb {C}}^*)^2/({\mathbb {Z}}/2{\mathbb {Z}}), \qquad \rho \mapsto (M,L), \end{aligned}$$
where \(\{M,1/M\}\) (resp., \(\{L,1/L\}\)) are the eigenvalues of \(\rho (\mu )\) (resp., \(\rho (\lambda )\)) where \((\mu ,\lambda )\) is a meridian-longitude pair on \(\partial M\). \(X_M\) and \(X_{\partial M}\) are affine varieties and the restriction map \(X_M \longrightarrow X_{\partial M}\) is algebraic. The Zariski closure of its image lifted to \(({\mathbb {C}}^*)^2\), and after removing any 0-dimensional components is a one-dimensional plane curve with defining polynomial \(A_K(M,L)\) [6]. This polynomial plays an important role in the hyperbolic geometry of the knot complement. We are now ready to formulate the AJ Conjecture [16]; see also [21]. Let us say that two polynomials \(P(M,L)=_M Q(M,L)\) are essentially equal if their irreducible factors with positive L-degree are equal.

Conjecture 2.2

For all knots K, we have \(A_K(M^2,L,1)=_M A_K(M,L)\).

The AJ Conjecture was checked for the \(3_1\) and the \(4_1\) knots in [16]. It is known for most 2-bridge knots [45], for torus knots and for the pretzel knots of Sect. 4; see [46, 55].

From the point of view of physics, the AJ Conjecture is a consequence of the fact that quantization and the corresponding quantum field theory exists [14, 31].

3 The volume and modularity conjectures

3.1 The volume conjecture

The Kashaev invariant of a knot is a sequence of complex numbers defined by [42, 47]
$$\begin{aligned} \langle K \rangle _N={\hat{J}}_{K,N}(e(1/N)), \end{aligned}$$
where \(e(\alpha )=e^{2 \pi i \alpha }\). The Volume Conjecture concerns the exponential growth rate of the Kashaev invariant and states that
$$\begin{aligned} \lim _N \frac{1}{N} \log |\langle K \rangle _N|=\frac{\text {vol}(K)}{2\pi }, \end{aligned}$$
where \(\text {Vol(K)}\) is the volume of the hyperbolic pieces of the knot complement \(S^3\setminus K\) [54]. Among hyperbolic knots, the Volume Conjecture is known only for the \(4_1\) knot. Detailed computations are available in [49]. Refinements of the Volume Conjecture to all orders in N and generalizations were proposed by several authors [13, 17, 26, 28]. Although proofs are lacking, there appears to be a lot of structure in the asymptotics of the Kashaev invariant. In the next section, we will discuss a modularity conjecture of Zagier and some numerical verification.

3.2 The modularity conjecture

Zagier considered the Galois invariant spreading of the Kashaev invariant on the set of complex roots of unity given by
$$\begin{aligned} \phi _K: {\mathbb {Q}}/{\mathbb {Z}}\longrightarrow {\mathbb {C}}, \qquad \phi _K\left( \frac{a}{c}\right) ={\hat{J}}_{K,c}\left( e\left( \frac{a}{c}\right) \right) , \end{aligned}$$
where \((a,c)=1\) and \(c>0\). The above formula works even when a and c are not coprime due to a symmetry of the colored Jones polynomial [36]. \(\phi _K\) determines \(\langle K \rangle \) and conversely is determined by \(\langle K \rangle \) via Galois invariance.

Let \(\gamma =\left( \begin{matrix} a &{} b \\ c &{} d \end{matrix}\right) \in \mathrm {SL}(2,{\mathbb {Z}})\) and \(\alpha =a/c\) and \(\hbar =2 \pi i/(X+d/c)\) where \(X \longrightarrow +\infty \) with bounded denominators. Let \(\phi =\phi _K\) denote the extended Kashaev invariant of a hyperbolic knot K and let \(F \subset {\mathbb {C}}\) denote the invariant trace field of \(M=S^3\setminus K\) [48]. Let \(C(M) \in {\mathbb {C}}/(4 \pi ^2 {\mathbb {Z}})\) denote the complex Chern–Simons invariant of M [35, 50]. The next conjecture was formulated by Zagier.

Conjecture 3.1

[65] With the above conventions, there exist \(\Delta (\alpha ) \in {\mathbb {C}}\) with \(\Delta (\alpha )^{2c} \in F(\epsilon (\alpha ))\) and \(A_j(\alpha ) \in F(e(\alpha ))\) such that
$$\begin{aligned} \frac{\phi (\gamma X)}{\phi (X)} \sim \left( \frac{2\pi }{\hbar }\right) ^{3/2} e^{C(M)/\hbar } \Delta (\alpha ) \sum _{j=0}^\infty A_j(\alpha ) \hbar ^j. \end{aligned}$$
(3)

When \(\gamma =\left( \begin{matrix} 1 &{} 0 \\ 1 &{} 1 \end{matrix}\right) \) and \(X=N-1\), and with the properly chosen orientation of M, the leading asymptotics of (3) together with the fact that \(\mathfrak {I}(C(M))=\text {vol}(M)\) gives the volume conjecture.

4 Computation of the non-commutative A-polynomial

As we will discuss below, the key to an effective computation the Kashaev invariant is a recursion for the colored Jones polynomial. Proving or guessing such a recursion is at least as hard as computing the A-polynomial of the knot. The A-polynomial is already unknown for several knots with 9 crossings. For an updated table of computations see [8]. The A-polynomial is known for the 1-parameter families of twist knots \(K_p\) [39] and pretzel knots \(KP_p=(-2,3,3+2p)\) [29] depicted on the left and the right part of the following figure
where an integer m inside a box indicates the number of |m| half-twists, right-handed (if \(m>0\)) or left-handed (if \(m<0\)), according to the following figure
The non-commutative A-polynomial of the twist knots \(K_p\) was computed with a certificate by X. Sun and the author in [30] for \(p=-14,\dots ,15\). The data are available from

http://www.math.gatech.edu/~stavros/publications/twist.knot.data

The non-commutative A-polynomial of the pretzel knots \(KP_p=(-2,3,3+3p)\) was guessed by C. Koutschan and the author in [23] for \(p=-5,\dots ,5\). The guessing method used an a priori knowledge of the monomials of the recursion, together with computation of the colored Jones polynomial using the fusion formula, and exact but modular arithmetic and rational reconstruction. The data are available from

http://www.math.gatech.edu/~stavros/publications/pretzel.data

For instance, the recursion relation for the colored Jones polynomial f(n) of the \(5_2=(-2,3,-1)\) pretzel knot is given by
$$\begin{aligned} \begin{aligned}&b(q^n,q) -q^{9+7 n} (-1+q^n) (-1+q^{2+n}) (1+q^{2+n}) (-1+q^{5+2 n}) f(n)\\&\quad +q^{5+2 n} (-1+q^{1+n})^2 (1+q^{1+n})(-1+ q^{5+2 n}) (-1+q^{1+n}+q^{1+2 n}-q^{2+2 n}\\&\quad -q^{3+2 n}+q^{4+2 n} -q^{2+3 n}-q^{5+3 n}-2 q^{5+4 n}+q^{6+5 n}) f(1+n)\\&\quad -q (-1+q^{2+n})^2 (1+q^{2+n}) (-1+q^{1+2 n}) (-1+2 q^{2+n}+q^{2+2n}+q^{5+2 n}\\&\quad -q^{4+3 n}+q^{5+3 n}+q^{6+3 n}-q^{7+3 n}-q^{7+4 n}+q^{9+5 n}) \\&\quad \times f(2+n)-(-1+q^{1+n}) (1+q^{1+n})(-1+q^{3+n}) (-1+q^{1+2 n}) f(3+n)=0, \end{aligned} \end{aligned}$$
where
$$\begin{aligned} b(q^n,q)=q^{4+2 n} (1+q^{1+n}) (1+q^{2+n}) (-1+q^{1+2 n}) (-1+q^{3+2 n}) (-1+q^{5+2 n}). \end{aligned}$$
The recursion relation for the colored Jones polynomial f(n) of the \((-2,3,7)\) pretzel knot is given by
$$\begin{aligned} \begin{aligned}&b(q^n,q) -q^{224+55 n} (-1+q^n) (-1+q^{4+n}) (-1+q^{5+n}) f(n)\\&\quad +q^{218+45 n} (-1+q^{1+n})^3 (-1+q^{4+n}) (-1+q^{5+n}) f(1+n)\\&\quad +q^{204+36 n} (-1+q^{2+n})^2 (1+q^{2+n}+q^{3+n}) (-1+q^{5+n}) f(2+n)\\&\quad +(-1+q) q^{180+27 n} (1+q) (-1+q^{1+n}) (-1+q^{3+n})^2 (-1+q^{5+n}) f(3+n)\\&\quad -q^{149+18 n} (-1+q^{1+n}) (-1+q^{4+n})^2 (1+q+q^{4+n}) f(4+n)\\&\quad -q^{104+8 n} (-1+q^{1+n}) (-1+q^{2+n}) \\&\quad (-1+q^{5+n})^3 f(5+n)+q^{59} (-1+q^{1+n}) (-1+q^{2+n}) (-1+q^{6+n}) f(6+n)=0, \end{aligned} \end{aligned}$$
where
$$\begin{aligned} \begin{aligned} b(q^n,q)&= q^{84+5 n} (1-q^{1+n}-q^{2+n}+q^{3+2 n}-q^{16+3 n}+q^{17+4 n}+q^{18+4 n}\\&\quad -q^{19+5 n}-q^{26+5 n}+q^{27+6 n}+q^{28+6 n}+q^{31+6 n}-q^{29+7 n}-q^{32+7 n}\\&\quad -q^{33+7 n}-q^{36+7 n}+q^{34+8 n}+q^{37+8 n}+q^{38+8 n}-q^{39+9 n}+q^{45+9 n}-q^{46+10 n}\\&\quad -q^{47+10 n}+q^{49+10 n}+q^{48+11 n}-q^{50+11 n}-q^{51+11 n}-q^{54+11 n}\\&\quad +q^{52+12 n}+q^{55+12 n}+q^{56+12 n}-q^{57+13 n}-q^{62+13 n}+q^{63+14 n}+q^{64+14 n}\\&\quad -q^{66+14 n}+q^{67+14 n}-q^{65+15 n}+q^{67+15 n}-q^{69+15 n}+q^{71+15 n}-q^{69+16 n}\\&\quad +q^{70+16 n}-q^{72+16 n}-q^{75+17 n}-q^{78+17 n}+q^{76+18 n}+q^{79+18 n}-q^{83+19 n}+q^{85+19 n}\\&\quad +q^{84+20 n}-q^{86+20 n}+q^{88+20 n}-q^{89+21 n}+q^{91+21 n}-q^{96+22 n}-q^{93+23 n}\\&\quad +2 q^{98+24 n}-q^{99+25 n}-q^{108+26 n}-q^{107+27 n}+q^{109+27 n}+q^{108+28 n}-q^{110+28 n}\\&\quad +q^{112+28 n}-q^{113+29 n}+q^{115+29 n}+q^{112+30 n}+q^{115+30 n}-q^{117+31 n}-q^{120+31 n}\\&\quad -q^{117+32 n}+q^{118+32 n}-q^{120+32 n}-q^{119+33 n}+q^{121+33 n}-q^{123+33 n}\\&\quad +q^{125+33 n}+q^{123+34 n}+q^{124+34 n}-q^{126+34 n}+q^{127+34 n}-q^{123+35 n}-q^{128+35 n}\\&\quad +q^{124+36 n}+q^{127+36 n}+q^{128+36 n}+q^{126+37 n}-q^{128+37 n}-q^{129+37 n}-q^{132+37 n}\\&\quad -q^{130+38 n}-q^{131+38 n}+q^{133+38 n}-q^{129+39 n}+q^{135+39 n}+q^{130+40 n}+q^{133+40 n}\\&\quad +q^{134+40 n}-q^{131+41 n}-q^{134+41 n}-q^{135+41 n}-q^{138+41 n}+q^{135+42 n}\\&\quad +q^{136+42 n}+q^{139+42 n}-q^{133+43 n}-q^{140+43 n}+q^{137+44 n}+q^{138+44 n}-q^{142+45 n}\\&\quad +q^{135+46 n}-q^{139+47 n}-q^{140+47 n}+q^{144+48 n}). \end{aligned} \end{aligned}$$
The pretzel knots \(KP_p\) are interesting from many points of view. For every integer p, the knots in the pair \((KP_p, -KP_{-p})\) (where \(-K\) denotes the mirror of K)
  • are geometrically similar, in particular they are scissors congruent, have equal volume, equal invariant trace fields and their Chern–Simons invariant differs by a sixth root of unity,

  • their A-polynomials are equal up to a \(\mathrm {GL}(2,{\mathbb {Z}})\) transformation [29, Thm.1.4].

Yet, the colored Jones polynomials of \((KP_p, -KP_{-p})\) are different, and so are the Kashaev invariants and their asymptotics and even the term \(\Delta (0)\) in the modularity conjecture 3.1. An explanation of this difference is given in [11].

Zagier posed a question to compare the modularity conjecture for geometrically similar pairs of knots, which was a motivation for many of the computations in Sect. 5.2.

5 Numerical asymptotics and the modularity conjecture

5.1 Numerical computation of the Kashaev invariant

To numerically verify Conjecture 3.1, we need to compute the Kashaev invariant to several hundred digits when \(N=2000\) for instance. In this section, we discuss how to achieve this.

There are multidimensional R-matrix state sum formulas for the colored Jones polynomial \(J_{K,N}(q)\) where the number of summation points is given by a polynomial in N of degree the number of crossings of K minus 1 [25]. Unfortunately, this is not practical method even for the \(4_1\) knot.

An alternative way is to use fusion [7, 32, 43] which allows one to compute the colored Jones polynomial more efficiently at the cost that the summand is a rational function of q. For instance, the colored Jones polynomial of a 2-fusion knot can be computed in \(O(N^3)\) steps using [23, Thm.1.1]. This method works better, but it still has limitations.

A preferred method is to guess a non-trivial recursion relation for the colored Jones polynomial (see Sect. 4) and instead of using it to compute the colored Jones polynomial, differentiate sufficiently many times and numerically compute the Kashaev invariant. In the efforts to compute the Kashaev invariant of the \((-2,3,7)\) pretzel knot, Zagier and the author obtained the following lemma, of theoretical and practical use.

Lemma 5.1

The Kashaev invariant \(\langle K \rangle _N\) can be numerically computed in O(N) steps.

A computer implementation of Lemma 5.1 is available.

5.2 Numerical verification of the modularity conjecture

Given a sequence of complex number \((a_n)\) with an expected asymptotic expansion
$$\begin{aligned} a_n \sim \lambda ^n n^{\alpha } (\log n)^{\beta } \sum _{j=0}^\infty \frac{c_j}{n^j} \end{aligned}$$
how can one numerically compute \(\lambda \), \(\alpha \), \(\beta \) and several coefficients \(c_j\)? This is a well-known numerical analysis problem [5]. An acceleration method was proposed in [62, p. 954], which is also equivalent to the Richardson transform. For a detailed discussion of the acceleration method see [22, Sect. 5.2]. In favorable circumstances, the coefficients \(c_j\) are algebraic numbers, and a numerical approximation may lead to a guess for their exact value.

A concrete application of the acceleration method was given in the appendix of [32] where one deals with several \(\lambda \) of the same magnitude as well as \(\beta \ne 0\).

Numerical computations of the modularity conjecture for the \(4_1\) knot were obtained by Zagier around roots of unity of order at most 5, and extended to several other knots in [33, 34]. As a sample computation, we present here the numerical data for \(4_1\) at \(\alpha =0\), computed independently by Zagier and by the author. The values of \(A_k\) in the table below are known for \(k=0,\dots ,150\).
$$\begin{aligned} \phi _{4_1}(X)= & {} 3^{-1/4} X^{3/2} \exp (C X) \left( \sum _{k=0}^\infty \frac{A_k}{k! 12^k} h^k \right) ,\\ h= & {} A/X \qquad A=\frac{\pi }{3^{3/2}} \qquad C=\frac{1}{\pi }\mathrm {Li_2}(\exp (2 \pi i/3)). \end{aligned}$$

k

\(A_k\)

0

1

1

11

2

697

3

724351/5

4

278392949/5

5

244284791741/7

6

1140363907117019/35

7

212114205337147471/5

8

367362844229968131557/5

9

44921192873529779078383921/275

10

3174342130562495575602143407/7

11

699550295824437662808791404905733/455

12

14222388631469863165732695954913158931/2275

13

5255000379400316520126835457783180207189/175

14

4205484148170089347679282114854031908714273/25

15

16169753990012178960071991589211345955648397560689/14875

16

119390469635156067915857712883546381438702433035719259/14875

17

1116398659629170045249141261665722279335124967712466031771/16625

18

577848332864910742917664402961320978851712483384455237961760783/914375

19

319846552748355875800709448040314158316389207908663599738774271783/48125

20

5231928906534808949592180493209223573953671704750823173928629644538303/67375

21

158555526852538710030232989409745755243229196117995383665148878914255633279/158125

22

2661386877137722419622654464284260776124118194290229321508112749932818157692851/186875

23

1799843320784069980857785293171845353938670480452547724408088829842398128243496119/8125

24

1068857072910520399648906526268097479733304116402314182132962280539663178994210946666679/284375

25

1103859241471179233756315144007256315921064756325974253608584232519059319891369656495819559/15925

26

8481802219136492772128331064329634493104334830427943234564484404174312930211309557188151604709/6125

In addition, we present the numerical data for the \(5_2\) knot at \(\alpha =1/3\), computed in [34].
$$\begin{aligned}&\phi _{5_2}(X/(3X+1))/\phi _{5_2}(X) \sim e^{C/h} (2 \pi /h)^{3/2} \Delta (1/3) \left( \sum _{k=0}^\infty A_k(1/3) h^k \right) ,\\&h=(2 \pi i)/(X+1/3),\\&F={\mathbb {Q}}(\alpha ) \qquad \alpha ^3-\alpha ^2+1=0 \qquad \alpha =0.877\dots -0.744\dots i,\\&C=R(1-\alpha ^2) + 2 R(1-\alpha ) -\pi i \log (\alpha )+\pi ^2, \\&R(x)=\mathrm {Li}_2(x)+\frac{1}{2} \log x \log (1-x)-\frac{\pi ^2}{6},\\&[1-\alpha ^2]+2[1-\alpha ] \in \mathcal {B}(F),\\&-23 = \pi _1^2 \pi _2 \qquad \pi _1=3\alpha -2 \qquad \pi _2=3\alpha +1,\\&\pi _7=(\alpha ^2-1)\zeta _6-\alpha +1 \qquad \pi _{43}=2\alpha ^2-\alpha -\zeta _6,\\&\Delta (1/3) = e(-2/9) \pi _7 \frac{3 \sqrt{-3}}{\sqrt{\pi _1}}, \\&A_0(1/3) = \pi _7 \pi _{43}, \\&A_1(1/3) = \frac{-952 + 321 \alpha -873 \alpha ^2 + (1348 + 557 \alpha + 26 \alpha ^2)\zeta _6}{ \alpha ^5 \pi _1^3}. \end{aligned}$$
One may use the recursion relations [24] for the twisted colored Jones polynomial to expand the above computations around complex roots of unity [9].

6 Stability

6.1 Stability of a sequence of polynomials

The Slope Conjecture deals with the highest (or the lowest, if you take the mirror image) q-exponent of the colored Jones polynomial. In this section, we discuss what happens when we shift the colored Jones polynomial and place its lowest q-exponent to 0. Stability concerns the coefficients of the resulting sequence of polynomials in q. A weaker form of stability (0-stability, defined below) for the colored Jones polynomial of an alternating knot was conjectured by Dasbach and Lin and proven independently by Armond [2].

Stability was observed in some examples of alternating knots by Zagier, and conjectured by the author to hold for all knots, assuming that we restrict the sequence of colored Jones polynomials to suitable arithmetic progressions, dictated by the quasi-polynomial nature of its q-degree [18, 19]. Zagier asked about modular and asymptotic properties of the limiting q-series.

A proof of stability in full for all alternating links is given in [27]. Besides stability, this approach gives a generalized Nahm sum formula for the corresponding series, which in particular implies convergence in the open unit disk in the q-plane. The generalized Nahm sum formula comes with a computer implementation (using as input a planar diagram of a link) and allows the computation of several terms of the q-series as well as its asymptotics when q approaches radially from within the unit circle a complex root of unity. The Nahm sum formula is reminiscent to the cohomological Hall algebra of motivic Donaldson–Thomas invariants of Kontsevich-Soibelman [44] and may be related to recent work of Witten [60] and Dimofte-Gaiotto-Gukov [12].

Let
$$\begin{aligned} {\mathbb {Z}}((q ))=\left\{ \sum _{n \in {\mathbb {Z}}} a_n q^n \, | \, a_n=0, \, n \ll 0 \right\} \end{aligned}$$
denote the ring of power series in q with integer coefficients and bounded below minimum q-degree.

Definition 6.1

Fix a sequence \((f_n(q))\) of polynomials \(f_n(q) \in {\mathbb {Z}}[q]\). We say that \((f_n(q))\) is 0-stable if the following limit exists
$$\begin{aligned} \lim _n f_n(q)=\Phi _0(q) \in {\mathbb {Z}}[[q ]], \end{aligned}$$
i.e., for every natural number \(m\in {\mathbb {Z}}\), there exists a natural number n(m) such that the coefficient of \(q^m\) in \(f_n(q)\) is constant for all \(n>n(m)\).
We say that \((f_n(q))\) is stable if there exist elements \(\Phi _k(q) \in {\mathbb {Z}}((q))\) for \(k=0,1,2,\dots \) such that for every \(k \in {\mathbb {N}}\) we have
$$\begin{aligned} \lim _n \, q^{-n k}\left( f_n(q)-\sum _{j=0}^{k} q^{jn} \Phi _j(q)\right) =0 \in {\mathbb {Z}}((q)). \end{aligned}$$
We will denote by
$$\begin{aligned} F(x,q)=\sum _{k=0}^\infty \Phi _k(q) x^k \in {\mathbb {Z}}((q ))[[x]] \end{aligned}$$
the corresponding series associated with the stable sequence \((f_n(q))\).

Thus, a 0-stable sequence \(f_n(q) \in {\mathbb {Z}}[q]\) gives rise to a q-series \(\lim _n f_n(q) \in {\mathbb {Z}}[[q]]\). The q-series that come from the colored Jones polynomial are q-hypergeometric series of a special shape, i.e., they are generalized Nahm sums. The latter are introduced in the next section.

6.2 Generalized Nahm sums

In [51], Nahm studied q-hypergeometric series \(f(q) \in {\mathbb {Z}}[[q]]\) of the form
$$\begin{aligned} f(q) =\sum _{n_1,\dots ,n_r \ge 0} \frac{q^{\frac{1}{2} n^t \cdot A \cdot n + b \cdot n}}{ (q)_{n_1} \dots (q)_{n_r}} \end{aligned}$$
where A is a positive definite even integral symmetric matrix and \(b \in {\mathbb {Z}}^r\). Nahm sums appear in character formulas in Conformal Field Theory and define analytic functions in the complex unit disk \(|q|<1\) with interesting asymptotics at complex roots of unity, and with sometimes modular behavior. Examples of Nahm sums are the famous list of seven mysterious q-series of Ramanujan that are nearly modular (in modern terms, mock modular). For a detailed discussion, see [64]. Nahm sums give rise to elements of the Bloch group, which governs the leading radial asymptotics of f(q) as q approaches a complex root of unity. Nahm’s Conjecture concerns the modularity of a Nahm sum f(q) and was studied extensively by Zagier, Vlasenko-Zwegers and others [58, 63].
The limit of the colored Jones function of an alternating link leads us to consider generalized Nahm sums of the form
$$\begin{aligned} \Phi (q)=\sum _{ n \in C \cap {\mathbb {N}}^r} (-1)^{c \cdot n} \frac{q^{\frac{1}{2} n^t \cdot A \cdot n + b \cdot n}}{(q)_{n_1} \dots (q)_{n_r}}, \end{aligned}$$
(4)
where C is a rational polyhedral cone in \({\mathbb {R}}^r\), \(b, c \in {\mathbb {Z}}^r\) and A is a symmetric (possibly indefinite) matrix. We will say that the generalized Nahm sum (4) is regular if the function
$$\begin{aligned} n \in C \cap {\mathbb {N}}^r \mapsto \frac{1}{2} n^t \cdot A \cdot n + b \cdot n \end{aligned}$$
is proper and bounded below, where \(\mathrm {mindeg}_q\) denotes the minimum degree with respect to q. Regularity ensures that the series (4) is a well-defined element of the ring \({\mathbb {Z}}((q))\). In the remaining of the paper, the term Nahm sum will refer to a regular generalized Nahm sum.

6.3 Stability for alternating links

Let K denote an alternating link. The lowest monomial of \(J_{K,n}(q)\) has coefficient \(\pm 1\), and dividing \(J_{K,n+1}(q)\) by its lowest monomial gives a polynomial \(J^+_{K,n}(q) \in 1 + q {\mathbb {Z}}[q]\). We can now quote the main theorem of [27].

Theorem 6.2

[27] For every alternating link K, the sequence \((J^+_{K,n}(q))\) is stable and the corresponding limit \(F_K(x,q)\) can be effectively computed by a planar projection D of K. Moreover, \(F_K(0,q)=\Phi _{K,0}(q)\) is given by an explicit Nahm sum computed by D.

An illustration of the corresponding q-series \(\Phi _{K,0}(q)\) the knots \(3_1\), \(4_1\) and \(6_3\) is given in Sect. 6.4.

6.4 Computation of the q-series of alternating links

Given the generalized Nahm sum for \(\Phi _{K,0}(q)\), a multidimensional sum of as many variables as the number of crossings of K, one may try to identify the q-series \(\Phi _{K,0}(q)\) with a known one. In joint work with Zagier, we computed the first few terms of the corresponding series (an interesting and non-trivial task in itself) and guessed the answer for knots with a small number of crossings. The guesses are presented in the following table

K

\(c_-\)

\(c_+\)

\(\sigma \)

\(\Phi ^*_{K,0}(q)\)

\(\Phi _{K,0}(q)\)

\(3_1=-K_1\)

3

0

2

\(h_3\)

\(h_2\)

\(4_1=K_{-1}\)

2

2

0

\(h_3\)

\(h_3\)

\(5_1\)

5

0

4

\(h_5\)

\(h_2\)

\(5_2=K_2\)

0

5

\(-\)2

\(h_4\)

\(h_3\)

\(6_1=K_{-2}\)

4

2

0

\(h_3\)

\(h_5\)

\(6_2\)

4

2

2

\(h_3 h_4\)

\(h_3\)

\(6_3\)

3

3

0

\(h_3^2\)

\(h_3^2\)

\(7_1\)

7

0

6

\(h_7\)

\(h_2\)

\(7_2=K_3\)

0

7

\(-\)2

\(h_6\)

\(h_3\)

\(7_3\)

0

7

\(-\)4

\(h_4\)

\(h_5\)

\(7_4\)

0

7

\(-\)2

\((h_4)^2\)

\(h_3\)

\(7_5\)

7

0

4

\(h_3 h_4\)

\(h_4\)

\(7_6\)

5

2

2

\(h_3 h_4\)

\(h_3^2\)

\(7_7\)

3

4

0

\(h_3^3\)

\(h_3^2\)

\(8_1=K_{-3}\)

6

2

0

\(h_3\)

\(h_7\)

\(8_2\)

6

2

4

\(h_3 h_6\)

\(h_3\)

\(8_3\)

4

4

0

\(h_5\)

\(h_5\)

\(8_4\)

4

4

2

\(h_4 h_5\)

\(h_3\)

\(8_5\)

2

6

\(-\)4

\(h_3\)

???

\(K_p, p>0\)

0

\(2p+1\)

\(-\)2

\(h_{2p}^*\)

\(h_3\)

\(K_p, p<0\)

2|p|

2

0

\(h_3\)

\(h_{2|p|+1}\)

\(T(2,p),p>0\)

\(2p+1\)

0

2p

\(h_{2p+1}\)

1

where \(\sigma \) denotes the signature of a knot [53] and for a positive natural number b, \(h_b\) are the unary theta and false theta series
$$\begin{aligned} h_b(q)=\sum _{n \in {\mathbb {Z}}} \varepsilon _b(n) \, q^{\frac{b}{2} n(n+1)-n}, \end{aligned}$$
where
$$\begin{aligned} \varepsilon _b(n) ={\left\{ \begin{array}{ll} (-1)^n &{} \text {if } b \text { is odd}, \\ 1 &{} \hbox {if } b \hbox { is even and } n \ge 0, \\ -1 &{} \hbox {if } b \hbox { is even and } n<0. \end{array}\right. } \end{aligned}$$
Observe that
$$\begin{aligned} h_1(q)=0, \qquad h_2(q)=1, \qquad h_3(q)=(q)_\infty . \end{aligned}$$
In the above table, \(c_+\) (resp. \(c_-\)) denotes the number of positive (resp., negative) crossings of an alternating knot K, and \(\Phi ^*_{K,0}(q)=\Phi _{-K,0}(q)\) denotes the q-series of the mirror \(-K\) of K, and T(2, p) denotes the (2, p) torus knot.
Concretely, the above table predicts the following identities
$$\begin{aligned} (q)_\infty ^{-2}= & {} \sum _{a,b,c \ge 0} (-1)^{a} \frac{q^{\frac{3}{2} a^2 + a b + a c + b c+ \frac{1}{2} a+ b+c}}{(q)_{a} (q)_{b} (q)_{c} (q)_{a+b} (q)_{a+c}}, \\ (q)_\infty ^{-3}= & {} \sum _{\begin{array}{c} a,b,c,d,e \ge 0 \\ a+b=d+e \end{array}} (-1)^{b+d}\frac{q^{\frac{b^2}{2}+ \frac{d^2}{2}+b c + a c + a d+b e + \frac{a}{2}+c + \frac{e}{2} }}{(q)_{b+c}(q)_{a} (q)_{b} (q)_{c} (q)_{d} (q)_{e} (q)_{c+d}}, \\ (q)_\infty ^{-4}= & {} \sum _{\begin{array}{c} a,b,c,d,e,f \ge 0 \\ a+e \ge b, b+f \ge a \end{array}} (-1)^{a-b+e}\\&\times \frac{q^{\frac{a}{2}+\frac{3 a^2}{2}+\frac{b}{2}+\frac{b^2}{2}+c+a c+d+a d+c d+\frac{e}{2}+2 a e-2 b e+d e+\frac{3 e^2}{2}-a f+b f+f^2}}{(q)_a (q)_b (q)_c (q)_{a+c} (q)_d (q)_{a+d} (q)_e (q)_{a-b+e} (q)_{a-b+d+e} (q)_f (q)_{-a+b+f}} \end{aligned}$$
corresponding to the knots
Some of the identities of the above table have been consequently proven [1]. In particular this settles the (mock)-modularity properties of the series \(\Phi _{K,0}(q)\) for all but one knot. The q-series of the remaining knot \(8_5\) is given by an eight-dimensional Nahm sum
where \(S=S(a,b,c,d,e,f,g,h)\) is given by
$$\begin{aligned}&S= (-1)^{b+f} \\&\,\, \frac{q^{2 a+3 a^2-\frac{b}{2}-2 a b+\frac{3 b^2}{2}+c+a c+d+a d+c d+e+a e+d e+\frac{3 f}{2}+4 a f-4 b f+e f+\frac{5 f^2}{2}+g+a g-b g+e g+f g+h+a h-b h+f h+g h}}{ (q)_{a} (q)_{b} (q)_{c} (q)_{d} (q)_{e} (q)_{f} (q)_{g} (q)_{h} (q)_{a+c} (q)_{a+d} (q)_{a+e} (q)_{a-b+f} (q)_{a-b+e+f} (q)_{a-b+f+g} (q)_{a-b+f+h}}. \end{aligned}$$
The first few terms of the series \(\Phi _{8_5,0}(q)\), which somewhat simplify when divided by \((q)_\infty \), are given by
$$\begin{aligned} \begin{aligned}&\Phi _{8_5,0}(q)/(q)_\infty = 1-q+q^2-q^4+q^5+q^6-q^8+2 q^{10}+q^{11}+q^{12}-q^{13}\\&\quad -2 q^{14}+2 q^{16}+3 q^{17}+2 q^{18}+q^{19}-3 q^{21}-2 q^{22}+q^{23}+4 q^{24}+4 q^{25}\\&\quad +5 q^{26}+3 q^{27}+q^{28}-2 q^{29}-3 q^{30}-3 q^{31}+5 q^{33}+8 q^{34}+8 q^{35}+8 q^{36}+6 q^{37}+3 q^{38}\\&\quad -2 q^{39}-5 q^{40}-6 q^{41}-q^{42}+2 q^{43}+9 q^{44}+13 q^{45}+17 q^{46}+16 q^{47}+14 q^{48}+9 q^{49}\\&\quad +4 q^{50}-3 q^{51}-8 q^{52}-8 q^{53}-5 q^{54}+3 q^{55}+14 q^{56}+21 q^{57}+27 q^{58}+32 q^{59}+33 q^{60}\\&\quad +28 q^{61}+21 q^{62}+11 q^{63}+q^{64}-9 q^{65}-11 q^{66}\\&\quad -11 q^{67}-2 q^{68}+9 q^{69}+27q^{70}+40 q^{71}+56 q^{72}+60 q^{73}+65 q^{74}+62 q^{75}+54 q^{76}\\&\quad +39 q^{77}+23 q^{78}+4 q^{79}-9 q^{80}-16 q^{81}-14 q^{82}-3 q^{83}+16 q^{84}+40 q^{85}+67 q^{86}\\&\quad +92 q^{87}+114 q^{88}+129 q^{89}+135 q^{90}+127 q^{91}+115 q^{92}+92q^{93}\\&\quad +66 q^{94}+35 q^{95}+9 q^{96}-12 q^{97}-14 q^{98}-11 q^{99}+13 q^{100}+O(q)^{101}. \end{aligned} \end{aligned}$$
We were unable to identify \(\Phi _{8_5,0}(q)\) with a known q-series. Nor were we able to decide whether it is a mock-modular form [64]. It seems to us that \(8_5\) is not an exception, and that the mock-modularity of the q-series \(\Phi _{8_5,0}(q)\) is an open problem.

Question 6.3

Can one decide if a generalized Nahm sum is a mock-modular form?

7 Modularity and stability

Modularity and Stability are two important properties of quantum knot invariants. The Kashaev invariant \(\langle K \rangle \) and the q-series \(\Phi _{K,0}(q)\) of a knotted three-dimensional object have some common features, namely asymptotic expansions at roots of unity approached radially (for \(\Phi _{K,0}(q)\)) and on the unit circle (for \(\langle K \rangle \)), depicted in the following figure
The leading asymptotic expansions of \(\langle K \rangle \) and \(\Phi _{K,0}(q)\) are governed by elements of the Bloch group as is the case of the Kashaev invariant and also the case of the radial limits of Nahm sums [58]. In this section, we discuss a conjectural relation, discovered accidentally by Zagier and the author in the spring of 2011, between the asymptotics of \(\langle 4_1\rangle \) and \(\Phi _{6j,0}(q)\), where 6j is the q-6j symbol of the tetrahedron graph whose edges are colored with 2N [7, 32]
The evaluation of the above tetrahedron graph \(J^+_{6j,N}(q) \in 1+q{\mathbb {Z}}[q]\) is given explicitly by [7, 32]
$$\begin{aligned} J^+_{6j,N}(q) = \frac{1}{1-q} \sum _{n=0}^{N} (-1)^n \frac{ q^{\frac{3}{2}n^2+\frac{1}{2}n}}{(q)_n^3} \frac{(q)_{4N+1-n}}{(q)_n^3 (q)_{N-n}^4}. \end{aligned}$$
The sequence \((J^+_{6j,N}(q))\) is stable, and the corresponding series \(F_{6j}(x,q)\) is given by
$$\begin{aligned} F_{6j}(x,q)= \frac{1}{(1-q)(q)_\infty ^3} \sum _{n=0}^\infty (-1)^n \frac{ q^{\frac{3}{2}n^2+\frac{1}{2}n}}{(q)_n^3} \frac{(x q^{-n})_\infty ^4}{(x^4 q^{-n+1})_\infty } \in {\mathbb {Z}}((q))[[x]], \end{aligned}$$
where as usual \((x)_\infty =\prod _{k=0}^\infty (1-xq^k)\) and \((q)_n=\prod _{k=1}^n (1-q^k)\). In particular,
$$\begin{aligned} \lim _N J^+_{6j,N}(q)=\Phi _{6j,0}(q)= \frac{1}{(1-q)(q)_\infty ^3} \sum _{n=0}^\infty (-1)^n \frac{q^{\frac{3}{2} n^2 + \frac{1}{2}n}}{(q)_n^3}. \end{aligned}$$
Let
$$\begin{aligned} \phi _{6j,0}(q) = \frac{(q)^4_\infty }{1-q}\Phi _{6j,0}(q)= (q)_\infty \sum _{n=0}^\infty (-1)^n \frac{q^{\frac{3}{2} n^2 + \frac{1}{2}n}}{(q)_n^3}. \end{aligned}$$
The first few terms of \(\phi _{6j,0}(q)\) are given by
$$\begin{aligned} \begin{aligned}&\phi _{6j,0}(q) =1-q-2 q^2-2 q^3-2 q^4+q^6+5 q^7+7 q^8+11 q^9+13 q^{10}\\&\quad +16 q^{11}+14 q^{12}+14 q^{13}+8 q^{14}-12 q^{16}-26 q^{17}-46 q^{18}-66 q^{19}-90 q^{20}-114 q^{21}\\&\quad -135 q^{22}-155 q^{23}-169 q^{24}-174 q^{25}-165 q^{26}-147 q^{27}-105 q^{28}-48 q^{29}+37 q^{30}\\&\quad +142 q^{31}+280 q^{32}+435 q^{33}+627 q^{34}+828 q^{35}+1060 q^{36}+O(q)^{37}. \end{aligned} \end{aligned}$$
The next conjecture which combines stability and modularity of two knotted objects has been numerically checked around complex roots of unity of order at most 3.

Conjecture 7.1

As \(X \longrightarrow +\infty \) with bounded denominator, we have
$$\begin{aligned} \phi _{6j,0}(e^{-1/X}) = \phi _{4_1}(X)/X^{1/2} + \overline{\phi _{4_1}(-{\bar{X}})/(- {\bar{X}})^{1/2}}. \end{aligned}$$

Declarations

Acknowledgements

The author was supported in part by NSF. To Don Zagier, with admiration.

Ethics approval and consent to participate

Not applicable.

Authors’ Affiliations

(1)
School of Mathematics, Georgia Institute of Technology, Atlanta, USA

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