Super Markov Numbers and Signed Double Dimer Covers

We provide a superalgebraic analogue of Markov numbers, which are defined as the Grassmann integer solutions to the equation $x^2+y^2+z^2+(xy+yz+xz)\epsilon =3(1+\epsilon )xyz$, as well as applications to the Decorated Super Teichmüller spaces associated to the once-punctured torus and certain annuli. We conclude with further directions for study.