Published paper
Abstract
We introduce a notion of -deformed rational numbers and -deformed continued fractions. A -deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the -deformed Pascal identity, with the Farey graph replacing Pascal's triangle. The coefficients count quiver subrepresentations, and the paper develops total positivity, a -deformed Farey graph, matrix presentations, -continuants, and a relation to Jones polynomials of rational knots.
Role in dependence graphs
Proof-critical source
On -real and -complex numbers
This paper is included only for the following marked statements:
- Theorems 1 and 2 · Forum Math. Sigma 8 (2020), VOR statements pp. 5-7; proof chain pp. 29-32Identifies the regular and negative continued-fraction definitions and gives positivity and strict monotonicity for finite -rationals.
- Appendix A, Propositions A.1-A.2 · Forum Math. Sigma 8 (2020), VOR pp. 44-45Identifies the rational-link Jones polynomial with the -rational numerator and supplies the explicit Burau matrices used at the endpoint.
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Exact reviewed source
Open Cambridge version of record · Forum of Mathematics, Sigma 8 (2020), e13
Sophie Morier-Genoud and Valentin Ovsienko. -deformed rationals and -continued fractions. Forum of Mathematics, Sigma 8 (2020), e13.
Open audited source ↗01Statements2 reported findingsNo statement issues found
The regular and negative continued-fraction definitions of the -deformed rational agree, its numerator and denominator have positive coefficients, the stated Farey and matrix recursions hold, the value is strictly increasing on positive rationals for , and the advertised relation to rational-link Jones polynomials is correct. A left/right matrix-multiplication word in Appendix A is a uniquely repairable typo, not a false formula.
Theorems 1 and 2No issue found
Well-definedness, positivity, and strict monotonicity have the advertised scope
Version-of-record pages 5–7 · Theorems 1–2
The regular-to-negative conversion formula, together with Proposition 4.9, identifies the two q-deformed convergent matrices up to the displayed scalar and final elementary matrix; their first-column ratios therefore agree. The triangulation/Farey recursion produces numerator and denominator polynomials with nonnegative coefficients. Theorem 2 expresses the cross-difference for any ordered pair of rationals as a polynomial with positive coefficients, so evaluation at proves the strict ordering statement used downstream.
Cambridge version of record ↗Appendix A, Propositions A.1–A.2Correct after a typographical repair
The rational-link Jones numerator formula is correct
Version-of-record pages 43–45 · Appendix A
After translating between the regular continued-fraction braid word and the -matrix product, the normalized Jones polynomial is the numerator polynomial specified in Proposition A.2, up to the stated monomial convention. The Lee-Schiffler continued-fraction formula independently matches the same recursion and initial case. The single rescaling typo described under proofs does not alter the displayed matrix or determinant.
02Proofs3 reported findingsNo proof issues found
The Farey-triangulation recurrence, matrix products, total-positivity deductions, and rational-link appendix form complete proofs of the marked claims. Appendix A.2 calls a column rescaling left multiplication; changing it to right multiplication is the unique dimensionally and algebraically consistent repair, and every subsequent formula already uses the repaired operation.
Farey and continuant proofsNo issue found in the reviewed proof
The matrix and positivity identities close the agreement and monotonicity arguments
Version-of-record pages 5–7 and 29–32 · Theorems 1–2 and Sections 4.6–4.7
Proposition 4.9 converts the regular matrix product to the negative one and, with Proposition 4.3, proves Theorem 1. For Theorem 2, Lemma 4.10 identifies the cross-difference of an arbitrary ordered pair with a power of times the numerator of an intermediate negative q-rational. Proposition 1.3 makes that numerator coefficientwise positive, proving the polynomial identity and strict ordering without a limiting argument.
Cambridge version of record ↗Appendix A.2Suspected typo · an apparently sound repair
The printed side of a diagonal multiplication is reversed
Version-of-record page 45 · proof of Proposition A.2
The proof says that rescaling the columns of a matrix is accomplished by multiplying on the left by the diagonal matrix . Column rescaling is , multiplication on the right. The displayed entries, the determinant calculation, and the Jones-polynomial identification all correspond to right multiplication, so replacing “left” by “right” repairs the sentence uniquely.
Appendix A source chainNo issue found in the reviewed proof
The imported continued-fraction Jones identity is independently checkable
Version-of-record pages 43–45; Lee-Schiffler arXiv:1710.08063v2 pages 28–30
Lee-Schiffler's Theorem 7.4 has the exact numerator recurrence required in Appendix A. For this selected identity, one can also induct directly from Lee-Schiffler's self-contained skein recursion in Theorem 5.5 and the ordinary continuant recurrence, so the conclusion does not depend on an unchecked cluster-submodule enumeration.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.