Quantum knot invariants
 Stavros Garoufalidis^{1}Email author
https://doi.org/10.1007/s4068701801273
© SpringerNature 2018
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
Mathematics Subject Classification
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 25yearold 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 above relations make clear that the colored Jones polynomial of a knot encodes the Jones polynomials of the knot and its 0framed 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 3space 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 \(\Delta _K(1)=1\).
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].
Conjecture 2.1
2.3 The colored Jones polynomial and the \(\mathrm {PSL}(2,{\mathbb {C}})\) character variety
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 2bridge 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
3.2 The modularity conjecture
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
When \(\gamma =\left( \begin{matrix} 1 &{} 0 \\ 1 &{} 1 \end{matrix}\right) \) and \(X=N1\), 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 noncommutative Apolynomial
http://www.math.gatech.edu/~stavros/publications/twist.knot.data
The noncommutative Apolynomial 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

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 Apolynomials are equal up to a \(\mathrm {GL}(2,{\mathbb {Z}})\) transformation [29, Thm.1.4].
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 Rmatrix 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 2fusion 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 nontrivial 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
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 \(\beta \ne 0\).
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 
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) qexponent of the colored Jones polynomial. In this section, we discuss what happens when we shift the colored Jones polynomial and place its lowest qexponent to 0. Stability concerns the coefficients of the resulting sequence of polynomials in q. A weaker form of stability (0stability, 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 quasipolynomial nature of its qdegree [18, 19]. Zagier asked about modular and asymptotic properties of the limiting qseries.
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 qplane. 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 qseries 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 KontsevichSoibelman [44] and may be related to recent work of Witten [60] and DimofteGaiottoGukov [12].
Definition 6.1
Thus, a 0stable sequence \(f_n(q) \in {\mathbb {Z}}[q]\) gives rise to a qseries \(\lim _n f_n(q) \in {\mathbb {Z}}[[q]]\). The qseries that come from the colored Jones polynomial are qhypergeometric 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
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 qseries \(\Phi _{K,0}(q)\) the knots \(3_1\), \(4_1\) and \(6_3\) is given in Sect. 6.4.
6.4 Computation of the qseries of alternating links
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  \(h_{2p}^*\)  \(h_3\) 
\(K_p, p<0\)  2p  2  0  \(h_3\)  \(h_{2p+1}\) 
\(T(2,p),p>0\)  \(2p+1\)  0  2p  \(h_{2p+1}\)  1 
Question 6.3
Can one decide if a generalized Nahm sum is a mockmodular form?
7 Modularity and stability
Conjecture 7.1
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
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