Perfect matching problems in cluster algebras and number theory

Abstract

This paper is a slightly extended version of the talk I gave at the Open Problems in Algebraic Combinatorics conference at the University of Minnesota in May 2022. We introduce two strict order relations on lattice paths and formulate several open problems. The topic is related to Markov numbers, the Lagrange spectrum, snake graphs and the cluster algebra of the once punctured torus. Our lattice paths are required to proceed by North and East steps and never go over the diagonal. To define the order relations, we first construct a snake graph $\mathcal{G}(\omega)$ and a band graph $\overline{\mathcal{G}(\omega)}$ for every such lattice path $\omega$. The first order relation $<_M$ is given by the number of perfect matchings of the snake graphs. The second order relation $<_L$ is given by the Lagrange number of a quadratic irrational associated to the band graph.