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Perturbative Knot Invariants via Quantum Cluster Algebras

Boudewijn Bosch

Abstract

A perturbative expansion of knot invariants is derived using quantum cluster algebras. By interpreting the $R$-matrix of $U_q(\mathfrak{sl}_2)$ as a cluster transformation and introducing an auxiliary parameter $ε$, we derive a perturbed $R$-matrix expressed in terms of Heisenberg algebra generators arising from the representation theory of the quantum cluster algebra. The resulting knot invariant has a zeroth-order term equal to $Δ_K(T)^{-1}$, the reciprocal of the Alexander polynomial, while higher-order terms in $ε$ produce perturbed Alexander polynomials. The construction combines the Schrödinger representation of the quantum torus algebra with cluster mutation combinatorics and is illustrated with a \textit{Mathematica} implementation and explicit examples.

Perturbative Knot Invariants via Quantum Cluster Algebras

Abstract

A perturbative expansion of knot invariants is derived using quantum cluster algebras. By interpreting the -matrix of as a cluster transformation and introducing an auxiliary parameter , we derive a perturbed -matrix expressed in terms of Heisenberg algebra generators arising from the representation theory of the quantum cluster algebra. The resulting knot invariant has a zeroth-order term equal to , the reciprocal of the Alexander polynomial, while higher-order terms in produce perturbed Alexander polynomials. The construction combines the Schrödinger representation of the quantum torus algebra with cluster mutation combinatorics and is illustrated with a \textit{Mathematica} implementation and explicit examples.
Paper Structure (12 sections, 9 theorems, 59 equations, 3 figures)

This paper contains 12 sections, 9 theorems, 59 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. The Schrödinger representation
  4. Symplectic vector spaces
  5. The Heisenberg group and the Schrödinger representation
  6. The Heisenberg algebra
  7. Cluster algebras
  8. Definitions
  9. Braiding operator
  10. The Alexander polynomial arising from cluster algebras
  11. Quantization
  12. Quantum cluster algebras

Key Result

Theorem 1.1

For a suitable cluster algebra with a distinguished choice of cluster variables, the mutation sequence associated with a triangulation of the complement of a knot $\mathcal{K}$ recovers the Alexander polynomial of $\mathcal{K}$ at the classical level; and after quantization and $\epsilon$-expansion,

Figures (3)

  • Figure 1: A quiver mutation in the direction $b$.
  • Figure 2: A triangulation of a 2-punctured disk
  • Figure 3: Half Dehn-twist for the 2-punctured disk.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 2.1: Stone--Von Neumann theorem
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Proposition 3.9
  • Proposition 3.10
  • Theorem 3.11
  • ...and 8 more