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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/

###### Custom metadata
24 pages, 65 figures
math.GT math.CO