On the Colored Jones Polynomial and the Kashaev invariant

2005/03/15 by Vu Huynh, Huynh, Vu, Thang T. Q. Le +1
Mathematics · #57M25 #FOS: Mathematics #Geometric Topology (math.GT) #Quantum Algebra (math.QA) #math.GT #math.QA #msc:57M25

paper · pdf · doi:10.48550/arxiv.math/0503296

14 pages

arxiv created 2005/03/15 · arxiv updated 2009/12/01

Abstract

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.

Related

• The colored Jones polynomials and the simplicial volume of a knot
• Quantum Knot Invariants
• The descendant colored Jones polynomials
• A non-commutative formula for the colored Jones function
• A determinant formula of the Jones polynomial for a family of braids