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2003 Volume Conjecture and Asymptotic Expansion of q-Series
Kazuhiro Hikami
Experiment. Math. 12(3): 319-338 (2003).

Abstract

We consider the "volume conjecture,'' which states that an asymptotic limit of Kashaev's invariant (or, the colored Jones type invariant) of knot {\small $\mathcal{K}$} gives the hyperbolic volume of the complement of knot {\small $\mathcal{K}$}. In the first part, we analytically study an asymptotic behavior of the invariant for the torus knot, and propose identities concerning an asymptotic expansion of q-series which reduces to the invariant with q being the N-th root of unity. This is a generalization of an identity recently studied by Zagier. In the second part, we show that "volume conjecture'' is numerically supported for hyperbolic knots and links (knots up to 6-crossing, Whitehead link, and Borromean rings).

Citation

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Kazuhiro Hikami. "Volume Conjecture and Asymptotic Expansion of q-Series." Experiment. Math. 12 (3) 319 - 338, 2003.

Information

Published: 2003
First available in Project Euclid: 15 June 2004

zbMATH: 1073.57006
MathSciNet: MR2034396

Subjects:
Primary: 11B65 , 52M27
Secondary: 11Mxx , 57M50

Keywords: hyperbolic volume , Jones polynomial , Rogers-Ramanujan identity

Rights: Copyright © 2003 A K Peters, Ltd.

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Vol.12 • No. 3 • 2003
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