Oriented posets, Rank Matrices and q-deformed Markov Numbers

We define oriented posets with correpsonding rank matrices, where linking two posets by an edge corresponds to matrix multiplication. In particular, linking chains via this method gives us fence posets, and taking traces gives us circular fence posets. As an application, we give a combinatorial model for $q$-deformed Markov numbers. We also resolve a conjecture of Leclere and Morier-Genoud and give several identities between circular rank polynomials.