The Large-Color Expansion Derived from the Universal Invariant

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.

💡 Research Summary

The paper establishes a direct derivation of the large‑color (Melvin–Morton–Rozansky) expansion of the colored Jones polynomial from the universal invariant associated with the Hopf algebra 𝔇, as introduced by Bar‑Natan and Van der Veen. The colored Jones polynomial Jₙ(K) is a quantum invariant obtained from the n‑dimensional irreducible representation of the quantum group U_q(sl₂). Historically, Melvin and Morton conjectured, and later Bar‑Natan–Garoufalidis proved, a relationship between Jₙ(K) and the Alexander polynomial Δ_K(T). Rozansky subsequently showed that Jₙ(K) admits an expansion \