The $q$-analog of the Markoff injectivity conjecture over the language of a balanced sequence

Abstract

The Markoff injectivity conjecture states that w → µ(w) 12 is injective on the set of Christoffel words where µ : {0, 1} * → SL 2 (Z) is a certain homomorphism and M 12 is the entry above the diagonal of a 2 × 2 matrix M . Recently, Leclere and Morier-Genoud (2021) proposed a q-analog µ q of µ such that µ q (w) 12 | q=1 = µ(w) 12 is the Markoff number associated to the Christoffel word w when evaluated at q = 1. We show that there exists an order < radix on {0, 1} * such that for every balanced sequence s ∈ {0, 1} Z and for all factors u, v in the language of s with u < radix v, the difference µ q (v) 12 − µ q (u) 12 is a nonzero polynomial of indeterminate q with nonnegative integer coefficients. Therefore, the map u → µ q (u) 12 is injective over the language of a balanced sequence. The proof uses an equivalence between balanced sequences satisfying some Markoff property and indistinguishable asymptotic pairs.

Quantum Continuants, Quantum Rotundus and Triangulations of Annuli

“…The subject has led to further developments in various directions. Notably there are established links with knots invariants [22], [18], the modular group and the Picard group [20], [33], combinatorics of posets [23], [30], [31], Markov numbers and Markov-Hurwitz approximation theory [9], [17], [19], [21], geometry of Grassmannians [32], triangulated categories [1].…”

“…3. The Farey triangulation T r s described in §3.3 correspond to the triangulated polygon (19) with parameters (c 1 , . .…”

Quantum continuants, quantum rotundus and triangulations of annuli

“…The subject has led to further developments in various directions. Notably there are established links with knots invariants [22], [18], the modular group and the Picard group [20], [33], combinatorics of posets [23], [30], [31], Markov numbers and Markov-Hurwitz approximation theory [9], [17], [19], [21], geometry of Grassmannians [32], triangulated categories [1].…”

A q-analog of the Markoff injectivity conjecture holds

“…In [9], a q-analog of the Markoff Injectivity Conjecture was considered based on the map w → µ q (w) 12 . It was proved that the map is injective over the language of any fixed Christoffel word, extending a result proved when q = 1 [10].…”