Torus knot state asymptotics

TL;DR

This paper analyzes the asymptotic behavior of torus knot states within Chern-Simons theory, linking quantum invariants to classical topological quantities, and confirms the Witten asymptotics conjecture for these knots.

Contribution

It computes the asymptotics of torus knot states using classical invariants and proves the microsupport inclusion, advancing understanding of quantum knot invariants in topological quantum field theory.

Findings

Asymptotic formulas involving Alexander polynomial, Reidemeister torsion, and Chern-Simons invariant.

Proof of the microsupport inclusion in the character manifold.

Verification of the Witten asymptotics conjecture for torus knots.

Abstract

The state of a knot is defined in the realm of Chern-Simons topological quantum field theory as a holomorphic section on the SU(2) character manifold of the peripheral torus. We compute the asymptotics of the torus knot states in terms of the Alexander polynomial, the Reidemeister torsion and the Chern-Simons invariant. We also prove that the microsupport of the torus knot state is included in the character manifold of the knot exterior. As a corollary we deduce the Witten asymptotics conjecture for the Dehn filling of the torus knots and asymptotic expansions for the colored Jones polynomials.