Volume conjecture and asymptotic expansion of q-series

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

We consider the "volume conjecture," which states that an asymptotic limit of Kashaev's invariant (or, the colored Jones type invariant) of knot K gives the hyperbolic volume of the complement of knot 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).

Original languageEnglish
Pages (from-to)319-337
Number of pages19
JournalExperimental Mathematics
Volume12
Issue number3
DOIs
Publication statusPublished - Jan 1 2003
Externally publishedYes

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Q-series
Asymptotic Expansion
Knot
Invariant
Borromean rings
Hyperbolic Volume
Hyperbolic Knot
Torus knot
Asymptotic Limit
Roots of Unity
Complement
Asymptotic Behavior

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Volume conjecture and asymptotic expansion of q-series. / Hikami, Kazuhiro.

In: Experimental Mathematics, Vol. 12, No. 3, 01.01.2003, p. 319-337.

Research output: Contribution to journalArticle

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