Alternating knots, planar graphs and q-series

Stavros Garoufalidis; Thao Vuong

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arXiv · 1304.1071

Alternating knots, planar graphs and q-series

Stavros Garoufalidis·Thao Vuong

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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 $q$-hypergeometric 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. Updated tables of data.

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Stavros Garoufalidis, Thao Vuong. 2013-04-03. Alternating knots, planar graphs and q-series. https://arxiv.org/abs/1304.1071

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