Asymptotic Behavior of Colored HOMFLY Polynomial of Figure Eight Knot

TL;DR

This paper studies the asymptotic properties of the colored HOMFLY polynomial for the figure eight knot, revealing connections to Chern-Simons invariants and Reidemeister torsion through asymptotic analysis.

Contribution

It establishes an asymptotic expansion for the colored HOMFLY polynomial of the figure eight knot, linking it to topological invariants like Chern-Simons invariants.

Findings

Asymptotic expansion for colored HOMFLY polynomial derived

Chern-Simons invariants obtained from asymptotics

Twisted Reidemeister torsion related to polynomial behavior

Abstract

In this paper we investigate the asymptotic behavior of the colored HOMFLY polynomial of the figure eight knot associated with the symmetric representation. We establish an analogous asymptotic expansion for the colored HOMFLY polynomial. From the asymptotic behavior we show that the Chern-Simons invariants and twisted Reidemeister torsion can be obtained with suitable modification of the case of colored Jones polynomial.