Higher $q$-Continued Fractions and Dimers on Band Graphs

Abstract

In this paper, we explore the theory of higher dimers on band graphs. First, we provide a combinatorial interpretation for the trace of the $q$-deformed higher continued fraction matrices, by showing that with respect to a $q$-weighting on edges, the trace gives the dimer partition function on the set of good higher dimers, which generalizes the notion of good perfect matchings. We also show that the set of good higher dimer covers form a distributive lattice with respect to face flips on square faces. Finally, we attempt to generalize the symmetry result on circular fence posets to the case of good higher dimers, by showing that the dimer partition on a certain family of band graphs are palindromic, in particular, through an approach fitting in the context of dimer theory.