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      Alternating knots, planar graphs and q-series

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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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          Journal
          03 April 2013
          2013-12-13
          Article
          1304.1071
          574ace69-ae72-4a9d-8020-5d3493bf2269

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          24 pages, 65 figures
          math.GT math.CO

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