{"paper":{"title":"Perturbed-Alexander Invariants via Quantum Cluster Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Interpreting the R-matrix of U_q(sl_2) as a cluster transformation produces a perturbative knot invariant whose leading term is the reciprocal of the Alexander polynomial.","cross_cats":[],"primary_cat":"math.GT","authors_text":"Boudewijn Bosch","submitted_at":"2026-03-16T19:42:07Z","abstract_excerpt":"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 "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"By interpreting the R-matrix of U_q(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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the R-matrix of U_q(sl_2) can be interpreted as a cluster transformation whose Schrödinger representation combined with cluster mutation combinatorics produces a well-defined perturbative knot invariant whose higher-order terms match the Bar-Natan–Van der Veen construction.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The construction yields a knot invariant whose leading term is the reciprocal of the Alexander polynomial, with higher-order terms in ε giving perturbed Alexander invariants via quantum cluster algebra techniques.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Interpreting the R-matrix of U_q(sl_2) as a cluster transformation produces a perturbative knot invariant whose leading term is the reciprocal of the Alexander polynomial.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4921babc2e2287e5235e420f42abc45ce6608b2e066e5da0bb851cf8ae859cd5"},"source":{"id":"2603.15859","kind":"arxiv","version":4},"verdict":{"id":"650fb52b-245e-41ed-826c-67e4f5ec9170","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T09:46:16.573028Z","strongest_claim":"By interpreting the R-matrix of U_q(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.","one_line_summary":"The construction yields a knot invariant whose leading term is the reciprocal of the Alexander polynomial, with higher-order terms in ε giving perturbed Alexander invariants via quantum cluster algebra techniques.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the R-matrix of U_q(sl_2) can be interpreted as a cluster transformation whose Schrödinger representation combined with cluster mutation combinatorics produces a well-defined perturbative knot invariant whose higher-order terms match the Bar-Natan–Van der Veen construction.","pith_extraction_headline":"Interpreting the R-matrix of U_q(sl_2) as a cluster transformation produces a perturbative knot invariant whose leading term is the reciprocal of the Alexander polynomial."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.15859/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"17e910c35cb68a3ce1dadbb0c8038bc1c4a3f791ce25e13f7a538e073c6951ec"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}