Volume conjecture and asymptotic expansion of \(q\)-series. (English) Zbl 1073.57006

Hikami, Kazuhiro

Exp. Math. 12, No. 3, 319-337 (2003).

Summary: We consider the “volume conjecture”, which states that an asymptotic limit of Kashaev’s invariant (or, the colored Jones type invariant) of a knot \({\mathcal K}\) gives the hyperbolic volume of the complement of \({\mathcal K}\). 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 the “volume conjecture” is numerically supported for hyperbolic knots and links (knots up to 6-crossings, Whitehead link, and Borromean rings).

MSC:

57M27	Invariants of knots and \(3\)-manifolds (MSC2010)
11B65	Binomial coefficients; factorials; \(q\)-identities
33D15	Basic hypergeometric functions in one variable, \({}_r\phi_s\)
57M50	General geometric structures on low-dimensional manifolds

Keywords:

Jones polynomial; Rogers-Ramanujan identity; hyperbolic volume

Software:

OEIS

References:

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