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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.15859 |
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Table of Contents:
- 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 invariants in line with the construction by Bar-Natan and Van der Veen. Our construction combines the Schrödinger representation of the quantum torus algebra with cluster mutation combinatorics and is illustrated with a Mathematica implementation and explicit examples.