An analytic version of the Melvin-Morton-Rozansky Conjecture

TL;DR

This paper provides an analytic formulation of the Melvin-Morton-Rozansky Conjecture by relating the growth rate of colored Jones polynomials at small angles to the loop expansion, advancing understanding of knot invariants.

Contribution

It establishes a precise analytic description of the polynomial growth rate of colored Jones polynomials, confirming a strong form of the Melvin-Morton-Rozansky Conjecture.

Findings

Identifies the growth rate with the loop expansion of the colored Jones function

Provides a strong analytic form of the Melvin-Morton-Rozansky Conjecture

Builds on previous results for small imaginary angles

Abstract

To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose $n$th term is the $n$th colored Jones polynomial. The Volume Conjecture for small angles states that the value of the $n$-th colored Jones polynomial at $e^{\a/n}$ is a sequence of complex numbers that grows subexponentially, for a fixed small complex angle $\a$. In an earlier publication, the authors proved the Volume Conjecture for small purely imaginary angles, using estimates of the cyclotomic expansion of a knot. The goal of the present paper is to identify the polynomial growth rate of the above sequence to all orders with the loop expansion of the colored Jones function. Among other things, this provides a strong analytic form of the Melvin-Morton-Rozansky conjecture. The resubmission corrects a misspelling of the first name of the second author.