Pith Number

pith:MU2XBXE6

pith:2026:MU2XBXE6TAPF6L3WSBXKTJP42Y

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Perturbed-Alexander Invariants via Quantum Cluster Algebras

Boudewijn Bosch

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.

arxiv:2603.15859 v4 · 2026-03-16 · math.GT

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

Formal links

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Receipt and verification

First computed	2026-05-21T01:04:24.803620Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

653570dc9e981e5f2f76906ea9a5fcd63bc50df017cf3053d6cab886ffe8086e

Aliases

arxiv: 2603.15859 · arxiv_version: 2603.15859v4 · doi: 10.48550/arxiv.2603.15859 · pith_short_12: MU2XBXE6TAPF · pith_short_16: MU2XBXE6TAPF6L3W · pith_short_8: MU2XBXE6

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/MU2XBXE6TAPF6L3WSBXKTJP42Y \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 653570dc9e981e5f2f76906ea9a5fcd63bc50df017cf3053d6cab886ffe8086e

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "e62574fb0cbef12e7f0e92ecc79f59e44cef1f52e0cf2ab418c587e7292abc84",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.GT",
    "submitted_at": "2026-03-16T19:42:07Z",
    "title_canon_sha256": "a55dd2efcda4a9f32651b5767f6247c6f7bc9424914d4ece323c4cbb15a166b9"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2603.15859",
    "kind": "arxiv",
    "version": 4
  }
}