Student combinatorics seminar: q-rational numbers and dimers

I will talk about a q-analogue of rational numbers introduced by Morier-Genoud and Ovsienko. Motivated by combinatorics of classic continued fractions in the context of triangulations and walks on the Farey graph, they assigned to each rational number r/s a rational function in q, R(q)/S(q), where R(q) and S(q) are coprime polynomials in q with positive integer coefficients. Based on a recent work of Ovsienko, we will interpret these polynomials in the context of perfect matchings (dimers) on snake graphs. This is a self-contained talk which will include necessary background on dimer models on planar graphs, such as the dimer lattice and the face polynomial.