A $q$-analog of the Markoff injectivity conjecture holds

Abstract

The elements of Markoff triples are given by coefficients in certain matrix products defined by Christoffel words, and the Markoff injectivity conjecture, a long-standing open problem (also known as the uniqueness conjecture), is then equivalent to injectivity on Christoffel words. A $q$-analog of these matrix products has been proposed recently, and we prove that injectivity on Christoffel words holds for this $q$-analog. The proof is based on the evaluation at $q=\exp (i\pi /3)$. Other roots of unity provide some information on the original problem, which corresponds to the case $q=1$. We also extend the problem to arbitrary words and provide a large family of pairs of words where injectivity does not hold.

v1: 9 pages, 2 figures. v2: 10 pages, 2 figures. v3: fixed few typos. v4: fixed an error in the statement of Theorem 3.2

References in corpus (2)

• Rank Polynomials of Fence Posets are Unimodal
• Towards quantized complex numbers: $q$-deformed Gaussian integers and the Picard group