FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2005, VOLUME 11, NUMBER 5, PAGES 57-78

**On the colored Jones polynomial and the Kashaev invariant**

Vu Huynh

T. T. Q. Lê

Abstract

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We express the colored Jones polynomial as the inverse of the quantum
determinant of a matrix with entries in the $q$-Weyl algebra of
$q$-operators,
evaluated at the trivial function (plus simple substitutions).
The Kashaev invariant is proved to be equal to another special
evaluation of the determinant.
We also discuss the similarity between our determinant formula of the
Kashaev invariant and the determinant formula of the hyperbolic volume
of knot complements, hoping it would lead to a proof of the volume
conjecture.

Location: http://mech.math.msu.su/~fpm/eng/k05/k055/k05505h.htm

Last modified: February 26, 2006