Alternating knots, planar graphs, and $$q$$ q -series

Abstract

Recent advances in Quantum Topology assign q-series to knots in at least three different ways. The q-series are given by generalized Nahm sums (i.e., special qhypergeometric sums) and have unknown modular and asymptotic properties. We give an efficient method to compute those q-series that come from planar graphs (i.e., reduced Tait graphs of alternating links) and compute several terms of those series for all graphs with at most 8 edges drawing several conclusions. In addition, we give a graph-theory proof of a theorem of Dasbach-Lin which identifies the coefficient of q k in those series for k = 0, 1, 2 in terms of polynomials on the number of vertices, edges and triangles of the graph. 10 4.

The tail of a quantum spin network

“…Recently, Garoufalidis and Vuong gave an algorithm for computing the tail of any alternating link [8]. In this section we apply the results we obtain in section 4 to study the tail of the color Jones polynomial.…”

“…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.…”

Rogers–Ramanujan type identities for alternating knots

“…For example, Hikami [14] considered (1.1) from the perspective of the colored Jones polynomial of torus knots while Armond and Dasbach [6] gave a skein-theoretic proof of (1.2). For similar identities related to false theta series, see [13] and for other connections between q-series and quantum invariants of knots, see [7]- [9], [11], [15] and [16].…”

Flag algebras and the stable coefficients of the Jones polynomial

“…Theorem 1.3 gives an expression for C k for k = 2, 3. To phrase our conjecture for C k for k = 4, 5, recall the notion of an irreducible planar graph from [GV15]. The latter is a planar graph which is not a vertex connected sum or an edge connected sum of planar graphs as in Figure 2.…”

“…Some lemmas from [GV15]. In this section we review the statements of some lemmas from [GV15] which we use for the proof of Theorem 1.3. The proofs of the three lemmas below can be found in [GV15, Sec.4].…”