The colored Jones polynomials and the Alexander polynomial of the figure-eight knot

Details The colored Jones polynomials and the Alexander polynomial of the figure-eight knot The colored Jones polynomials and the Alexander polynomial of the figure-eight knot

2005-02-20

arxiv.org/abs/math/0502428v1

JP J. Geom. Topol. 2 (2007), 249--269

Mathematics

Geometric Topology

13 pages

Scientific paper

The volume conjecture and its generalization state that the series of certain evaluations of the colored Jones polynomials of a knot would grow exponentially and its growth rate would be related to the volume of a three-manifold obtained by Dehn surgery along the knot. In this paper, we show that for the figure-eight knot the series converges in some cases and the limit equals the inverse of its Alexander polynomial.

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