Perturbed-Alexander Invariants via Quantum Cluster Algebras

TL;DR

This paper develops a perturbative expansion of knot invariants using quantum cluster algebras, connecting the R-matrix to cluster transformations and producing perturbed Alexander invariants.

Contribution

It introduces a novel approach to knot invariants by interpreting the R-matrix as a cluster transformation and deriving perturbed invariants via quantum cluster algebra techniques.

Findings

The zeroth-order term of the invariant equals the reciprocal of the Alexander polynomial.

Higher-order terms produce perturbed Alexander invariants consistent with prior constructions.

The method is illustrated with explicit examples and a Mathematica implementation.

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 $\epsilon$, 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 $\Delta_K(T)^{-1}$, the reciprocal of the Alexander polynomial, while higher-order terms in $\epsilon$ produce perturbed-Alexander invariants in line with the construction by Bar-Natan and Van der Veen. Our construction combines the Schr\"odinger representation of the quantum torus algebra with cluster mutation combinatorics and is illustrated with a Mathematica implementation and explicit examples.