The higher order terms in asymptotic expansion of color Jones polynomials

TL;DR

This paper develops an algorithm to compute higher order terms in the asymptotic expansion of the color Jones polynomial, connecting geometric invariants with quantum knot invariants and proposing a conjecture relating MMR expansion to the abelian branch of the A-polynomial.

Contribution

It introduces a general expansion framework for the color Jones polynomial, including an algorithm for higher order terms based on A-polynomial theory, and proposes a new conjecture linking MMR expansion to the abelian branch.

Findings

Algorithm for higher order asymptotic terms derived from A-polynomial

Conjecture relating MMR expansion to abelian branch of A-polynomial

Examples demonstrating the calculation of higher order terms

Abstract

Color Jones polynomial is one of the most important quantum invariants in knot theory. Finding the geometric information from the color Jones polynomial is an interesting topic. In this paper, we study the general expansion of color Jones polynomial which includes the volume conjecture expansion and the Melvin-Morton-Rozansky (MMR) expansion as two special cases. Following the recent works on SL(2,C) Chern-Simons theory, we present an algorithm to calculate the higher order terms in general asymptotic expansion of color Jones polynomial from the view of A-polynomial and noncommutative A-polynomial. Moreover, we conjecture that the MMR expansion corresponding to the abelian branch of A-polynomial. Lastly, we give some examples to illustrate how to calculate the higher order terms. These results support our conjecture.