Wug-snake graphs and Markov numbers of matrix semigroups

Abstract

Classically, Markov numbers are recovered as perfect matching numbers of domino snake graphs. We extend this correspondence by introducing weighted universal generalised snake graphs, or wug-snake graphs. These are weighted ordered bipartite graphs whose perfect matching sequences encode linear recurrences. To every wug-snake graph we associate a continuant matrix and prove that the determinant of this matrix equals the weighted perfect matching sum. We then introduce polyomino wug-tiles, bodies of wug-snake graphs that act linearly on state vectors. Every integer matrix admits a canonical polyomino wug-tile. Our main result identifies the wug-snake determinant of a tile representing a matrix $A$ with the Markov-Davenport form of $A$. Consequently, algebraic and geometric Markov numbers of matrices and matrix semigroups can be expressed as weighted perfect matching determinants. Further we define Frobenius maps for matrix semigroups and discuss examples recovering classical Markov numbers and higher-dimensional lattice realisations.