Published paper
Abstract
We establish a link between -deformed rational numbers and the classical Burau representation of the braid group . This gives a classification of faithful complex specializations in terms of the singular set of the -rationals and proves faithfulness outside the annulus .
Role in dependence graphs
Proof-critical source
On -real and -complex numbers
This paper is included only for the following marked statement:
- Theorem 1 · IMRN 2024 VOR pp. 8620-8624Equates a zero of a -rational numerator polynomial with nonfaithfulness of the specialized reduced Burau representation.
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Exact reviewed source
Open repository version of record · International Mathematics Research Notices 2024, no. 10, 8618–8627
Sophie Morier-Genoud, Valentin Ovsienko, and Alexander P. Veselov. Burau representation of braid groups and -rationals. International Mathematics Research Notices 2024, no. 10, 8618–8627.
Open audited source ↗01Statements2 reported findingsNo statement issues found
The singular-set characterization of nonfaithful specializations of the reduced Burau representation of and the resulting faithfulness region outside are correctly stated. The root condition is attached to the numerator polynomials of -rationals with the reciprocal and sign conventions needed for the specialization.
Theorem 1No issue found
Polynomial zeros and Burau nonfaithfulness are equivalent
Printed pages 8620–8624 · version of record
The projectivized Burau action is identified with the -deformed modular action. A nontrivial kernel word fixes the associated rational point, giving a numerator zero; conversely, a numerator zero makes two distinct modular words have the same specialized projective action. The scalar ambiguity is removed by the determinant/central-element calculation, so the conclusion concerns the linear Burau representation rather than only its projectivization.
IMRN version of record ↗Theorem 2No issue found
The annular faithfulness bound follows with reciprocal endpoints
Printed pages 8624–8626 · version of record
The numerator and denominator polynomials have reciprocal root symmetry, while the elementary coefficient bound excludes roots in the inner disk. Applying the symmetry excludes the exterior reciprocal disk, producing exactly the displayed annulus.
02Proofs2 reported findingsNo proof issues found
The modular-action identification, free-subgroup argument, scalar lift, and reciprocal-root estimate close the two main proofs. The proof-critical use of the Kurosh subgroup theorem is represented by a terminal book source, and the needed -rational facts are verified in the exact MO1 version of record.
Proof of Theorem 1No issue found in the reviewed proof
The free-subgroup and projective-lift steps rule out both directions of failure
Printed pages 8621–8624 · version of record
The relevant subgroup of the modular/free-product presentation is free by the Kurosh theorem. The -rational orbit map records its reduced words, so equality of specialized orbit values yields precisely a projective kernel relation. Tracking determinants and the image of the center then lifts the relation to the reduced Burau representation with no missing exceptional scalar case.
IMRN version of record ↗Proof of Theorem 2No issue found in the reviewed proof
The zero-free disk and reciprocity arguments are compatible
Printed pages 8624–8626 · version of record
Positivity of the -rational coefficients gives the uniform inner zero-free estimate. The involution on continued fractions exchanges numerator and denominator with the reciprocal-polynomial transformation, so every exterior root would produce a prohibited inner root. The endpoint arithmetic simplifies to as printed.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.