Abstract
We reveal an intimate connection between the quantum knot invariant for torus knot T(s,t) and the character of the minimal model M(s,t), where s and t are relatively prime integers. We show that Kashaev's invariant, i.e., the N-colored Jones polynomial at the Nth root of unity, coincides with the Eichler integral of the character.
Highlights
Kashaev defined a quantum knot invariant based on the quantum dilogarithm function [3], and made a conjecture [4] that a limit of his invariant coincides with the hyperbolic volume of the knot complement [5]
This suggests an intimate connection between the quantum invariant and the geometry
The Chern–Simons invariant is considered as an imaginary part of the hyperbolic volume, and the torus knot is supposed to have non-trivial Chern–Simons invariant
Summary
Note that Kashaev’s invariant was later identified with a specialization of the N-colored Jones polynomial at q being the N-th primitive root of unity [6]. We study Kashaev’s invariant K N for the torus knot K = T (s, t), where s and t are coprime. We shall show that the invariant exactly coincides with a limiting value of the Eichler integral of the character of the minimal model M(s, t) with q being the N-th root of unity.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
