Experimental Mathematics

Kashaev's Conjecture and the Chern-Simons Invariants of Knots and Links

Hitoshi Murakami, Jun Murakami, Miyuki Okamoto, Toshie Takata, and Yoshiyuki Yokota

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Abstract

R. M. Kashaev conjectured that the asymptotic behavior of the link invariant he introduced, which equals the colored Jones polynomial evaluated at a root of unity, determines the hyperbolic volume of any hyperbolic link complement. We observe numerically that for knots {\small $6_3$, $8_9$ and $8_{20}$} and for the Whitehead link, the colored Jones polynomials are related to the hyperbolic volumes and the Chern--Simons invariants and propose a complexification of Kashaev's conjecture.

Article information

Source
Experiment. Math., Volume 11, Issue 3 (2002), 427-435.

Dates
First available in Project Euclid: 9 July 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1057777432

Mathematical Reviews number (MathSciNet)
MR1959752

Zentralblatt MATH identifier
1117.57300

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M50: Geometric structures on low-dimensional manifolds
Secondary: 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15] 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23] 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37]

Keywords
Volume conjecture Kashaev's conjecture colored Jones polynomial Chern-Simons invariant volume

Citation

Murakami, Hitoshi; Murakami, Jun; Okamoto, Miyuki; Takata, Toshie; Yokota, Yoshiyuki. Kashaev's Conjecture and the Chern-Simons Invariants of Knots and Links. Experiment. Math. 11 (2002), no. 3, 427--435. https://projecteuclid.org/euclid.em/1057777432


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