The Large-Color Expansion Derived from the Universal Invariant

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.

arXiv comment21 pages (Corrected a few typos from the previous version, added the notion of XC-algebras, clarified multiple images, and improved wording)