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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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| _version_ | 1866918512607363072 |
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| author | Bosch, Boudewijn |
| author_facet | Bosch, Boudewijn |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15859 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Perturbed-Alexander Invariants via Quantum Cluster Algebras Bosch, Boudewijn Geometric Topology 57K14, 17B37, 13F60 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. |
| title | Perturbed-Alexander Invariants via Quantum Cluster Algebras |
| topic | Geometric Topology 57K14, 17B37, 13F60 |
| url | https://arxiv.org/abs/2603.15859 |