Coincidences of $q$-analogues of Markoff numbers outside the known families: an answer to a question of Labbé, Lapointe and Steiner

David Niedbala Giraudin

Coincidences of $q$ -analogues of Markoff numbers outside the known families: an answer to a question of Labbé, Lapointe and Steiner

Primarily AI-generated textHuman understanding: no partsmath.CO — Combinatorics math.NT — Number Theory

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

Labbé, Lapointe and Steiner proved that

w\mapsto\mu_q(w)_{12}

is injective on Christoffel words, gave two families of identities showing that it is not injective on

\{\mathtt{0},\mathtt{1}\}^*

, and asked whether every coincidence

\mu_q(x)_{12}=\mu_q(y)_{12}

comes from these two families. We show that it does not. The smallest counterexamples have length

24

; by an exhaustive computation, among words of length at most

32

there are exactly

50

classes of coincidences not explained by the two families,

10

of length

24

and

40

of length

30

. Each of them contains an instance of one of a few new identities, which we prove for a generic

2\times2

matrix in place of the matrix of a factor of the word. One of these identities belongs to an infinite family, indexed by

k\equiv1\pmod 4

, which we prove for every such

k

.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by independent AI reviews and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic: Claude Opus

References

S. Labbé, M. Lapointe and W. Steiner, A $q$ -analog of the Markoff injectivity conjecture holds, Algebraic Combinatorics 6 (2023), no. 6, 1677--1685; doi:10.5802/alco.322; arXiv:2212.09852 (latest version, with the corrected form of Theorem 3.2).arXiv DOI
L. Leclere and S. Morier-Genoud, $q$ -deformations in the modular group and of the real quadratic irrational numbers, Adv. in Appl. Math. 130 (2021), 102223; arXiv:2101.02953.arXiv
C. Procesi, The invariant theory of $n\times n$ matrices, Adv. Math. 19 (1976), 306--381.
C. Procesi, Computing with $2\times2$ matrices, J. Algebra 87 (1984), 342--359.
K. S. Sibirskii, Algebraic invariants of a system of matrices, Sibirsk. Mat. Zh. 9 (1968), 152--164.

Version history

v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 01, 2026