  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.
