Published paper
Abstract
The paper connects cluster algebras with knot theory through continued fractions and snake graphs. For a two-bridge link, the normalized Jones polynomial is a specialization of a cluster variable. In particular, it gives a direct continued-fraction numerator formula, degree and width formulas, extreme coefficients, and recursive Jones-polynomial identities.
Role in dependence graphs
Proof-critical source
On -real and -complex numbers
This paper is included only for the following marked statement:
- Theorem 1.2 / Theorem 7.4 · arXiv:1710.08063v2 fallback PDF pp. 3 and 29Expresses a rational-link Jones polynomial as the numerator of the corresponding continued fraction, the exact formula imported in MO1 Appendix A.
AI-generated audit
Audit summary
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Exact reviewed source
Pinned arXiv fallback · arXiv:1710.08063v2
Kyungyong Lee and Ralf Schiffler. Cluster algebras and Jones polynomials. Selecta Mathematica 25 (2019), article 58. Audited against arXiv:1710.08063v2 because the final publisher PDF is subscription-gated.
The Selecta Mathematica version of record is subscription-gated; the final arXiv version contains the numbered theorem and complete proof used downstream.
Open audited source ↗01Statements2 reported findingsNo statement issues found
The normalized Jones polynomial of a two-bridge link is correctly identified with a specialization of the continued-fraction snake-graph cluster variable. The direct numerator formula, degree and width formulas, recursive identities, and extreme-coefficient consequences have the stated orientation and normalization conventions. Theorem 1.2/Theorem 7.4 is the exact dependency used by MO1.
Theorem 1.2 / Theorem 7.4No issue found
The Jones polynomial is the stated continued-fraction numerator
Fallback PDF pages 3 and 28–30 · arXiv:1710.08063v2
With and the paper's leading-term normalization, the skein recursion for a terminal twist block agrees with the numerator-continuant recurrence. The one-block base case fixes the monomial and sign. Induction therefore gives the printed numerator formula for every positive continued fraction, and the even-expansion correspondence extends it to the stated two-bridge links.
Pinned arXiv fallback version 2 ↗Theorem 7.1, Corollary 7.2, and Theorem 7.6No issue found
Degree, width, and coefficient statements follow from the same expansion
Fallback PDF pages 27–32 · arXiv:1710.08063v2
The lowest and highest monomials are determined by the two extremal perfect matchings of the snake graph, and the next coefficients count the adjacent boundary choices. Their exponent difference matches the crossing/twist statistic printed in the width formula.
02Proofs2 reported findingsNo proof issues found
The even-to-positive continued-fraction snake-graph correspondence, Kauffman-bracket recursion, cluster-variable specialization, and extremal-matching counts form a consistent proof chain. For the exact downstream numerator identity, a direct induction from the paper's self-contained skein recursion independently bypasses the two external snake-graph enumeration citations.
Theorem 5.5 to Theorem 7.4No issue found in the reviewed proof
The skein recursion independently verifies the proof-critical formula
Fallback PDF page 16 and pages 27–29 · arXiv:1710.08063v2
Theorem 5.5 gives the two-term Jones recursion obtained by resolving the last twist region. Proposition 6.4 gives the continued-fraction numerator for the specialized F-polynomial, and Theorem 7.1 identifies that F-polynomial with the Jones polynomial by induction using the same recursion and one-block initial case. Theorem 7.4 is therefore the exact continued-fraction numerator identity needed by MO1; comparing the two recurrences also independently checks the marked formula without relying on an unexamined cluster-submodule enumeration.
Pinned arXiv fallback version 2 ↗Snake-graph correspondenceNo issue found in the reviewed proof
The local graph moves preserve the continued-fraction value and weights
Fallback PDF pages 7–13 and 20–28 · arXiv:1710.08063v2
The local replacement taking an even expansion to a positive expansion changes the tile sequence exactly as the elementary continued-fraction move. The induced perfect-matching weight transformation agrees with the coefficient specialization and , so the cluster and skein calculations use the same normalized Laurent polynomial.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.