Topological formula of the loop expansion of the colored Jones polynomials

TL;DR

This paper presents a topological formula for the loop expansion of colored Jones polynomials, linking quantum representations with homological representations, and providing a proof of the Melvin-Morton-Rozansky conjecture.

Contribution

It introduces a topological formula connecting quantum sl2 representations to homological representations, offering a new proof of a key conjecture and linking braid entropy to quantum invariants.

Findings

Provides a topological proof of the Melvin-Morton-Rozansky conjecture

Establishes a connection between braid entropy and quantum representations

Derives a topological formula for the loop expansion of colored Jones polynomials

Abstract

We give a topological formula of the loop expansion of the colored Jones polynomials by using identification of generic quantum sl2 representation with homological representations. This gives a direct topological proof of the Melvin-Morton-Rozansky conjecture, and a connection between entropy of braids and quantum representations.