Nahm sums, stability and the colored Jones polynomial

Abstract

Nahm sums are q-series of a special hypergeometric type that appear in character formulas in the conformal field theory, and give rise to elements of the Bloch group, and have interesting modularity properties. In our paper, we show how Nahm sums arise naturally in the quantum knot theory -we prove the stability of the coefficients of the colored Jones polynomial of an alternating link and present a Nahm sum formula for the resulting power series, defined in terms of a reduced diagram of the alternating link. The Nahm sum formula comes with a computer implementation, illustrated in numerous examples of proven or conjectural identities among q-series. MSC: Primary 57N10; Secondary 57M25.Keywords: Nahm sums; Colored Jones polynomial; Links; Stability; Modular forms; Mock-modular forms; q-holonomic sequence; q-series; Conformal field theory; Thin-thick decomposition BackgroundThe colored Jones polynomial of a link is a sequence of Laurent polynomials in one variable with integer coefficients. We prove in full a conjecture concerning the stability of the colored Jones polynomial for all alternating links.A weaker form of stability (zero stability, defined below) for the colored Jones polynomial of an alternating knot was conjectured by Dasbach and Lin. The zero stability is also proven independently by Armond for all adequate links [1], which include alternating links and closures of positive braids, see also [2]. The advantage of our approach is that it proves stability to all orders and gives explicit formulas (in the form of generalized Nahm sums) for the limiting series, which in particular implies convergence in the open unit disk in the q-plane and allow for the study of their redial asymptotics.Stability was observed in some examples by Zagier, and conjectured by the first 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 [3,4]. Zagier asked about modular and asymptotic properties of the limiting q-series. In a similar direction, Habiro asked about zero stability of the cyclotomic function of alternating links in [5].Our generalized Nahm sum formula comes with a computer implementation (using as input a planar diagram of a link), and allows the study of its asymptotics when q

The tail of a quantum spin network

“…The tail of this element can be seen to be Λ(q)(q 2 ; q) n . This tail was also computed by Garoufalidis and Le in [7]. …”

“…Higher stability of the coefficients of the colored Jones polynomial of an alternating link was shown by Garoufalidis and Le in [7]. Calculations of the tail of the colored Jones polynomial were done by a number of authors, see Armond and Dasbach [2], Garoufalidis and Le [7], and Hajij [9]. Recently, Garoufalidis and Vuong [8] have given an algorithm to compute the tails of the colored Jones polynomial of alternating links.…”

“…The existence of these two power series was conjectured by Dasbach and Lin [6] and was proven by Armond in [3]. Higher stability of the coefficients of the colored Jones polynomial of an alternating link was shown by Garoufalidis and Le in [7]. Calculations of the tail of the colored Jones polynomial were done by a number of authors, see Armond and Dasbach [2], Garoufalidis and Le [7], and Hajij [9].…”

“…[7]. Explicit calculations were done on the knot table to determine the tail of colored Jones polynomial of alternating links in [2].…”

The 3D index of an ideal triangulation and angle structures

“…By definition, I M is a generalized Nahm sum in the sense of [10], where the summation is over a lattice. · 1, it follows that the right hand side of the pentagon identity (3.5) is convergent in Z((q 1/2 )).…”