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The Large-Color Expansion Derived from the Universal Invariant

Boudewijn Bosch

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.

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 , as introduced by Bar-Natan and Van der Veen. We utilize a Mathematica implementation to compute the universal invariant up to a certain order for a given knot , allowing for experimental verification of our theoretical results.

Paper Structure

This paper contains 13 sections, 27 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. The universal invariant and twisting
  4. Tangles
  5. The universal invariant
  6. XC-algebras
  7. The twisted $R$-matrix
  8. Perturbed knot invariants
  9. A review of $U_q(\mathfrak{sl}_2)$ and the colored Jones polynomial
  10. The Drinfeld double $\mathfrak{D}_{\mathfrak{sl}_2}$
  11. The algebra $\mathbb{D}$
  12. The large-color expansion
  13. Experimental verification

Key Result

Theorem 1.1

The polynomials $\rho^{\mathcal{K}}_{1,0}$ and $P^{\mathcal{K}}_1$ are equal.

Theorems & Definitions (60)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Example 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • ...and 50 more