The Large-Color Expansion Derived from the Universal Invariant

TL;DR

This paper derives the large-color expansion of the colored Jones polynomial from the universal invariant associated with a specific Hopf algebra, providing a new algebraic perspective and computational tools for knot invariants.

Contribution

It introduces a method to derive the large-color expansion from the universal invariant of a Hopf algebra, supported by computational verification.

Findings

Derivation of the large-color expansion from the universal invariant.

Implementation of Mathematica code for invariant computation.

Experimental confirmation of theoretical results.

Abstract

The colored Jones polynomial associated to a knot admits an expansion of knot invariants known as the large-color expansion or Melvin-Morton-Rozansky expansion. We will show how this expansion can be derived from the universal invariant arising from a Hopf algebra $\mathbb{D}$, as introduced by Bar-Natan and Van der Veen. We utilize a Mathematica implementation to compute the universal invariant $\mathbf{Z}_{\mathbb{D}}(\mathcal{K})$ up to a certain order for a given knot $\mathcal{K}$, allowing for experimental verification of our theoretical results.