  Following McShane, we employ the stable norm on the homology of the modular
torus to investigate the Markov ordering on the set of relatively prime integer
pairs $(q,p)$ with $q\ge p\ge0$. Our main theorem is a characterization of
slopes along which the Markov ordering is monotone with respect to $q$,
confirming conjectures of Lee-Li-Rabideau-Schiffler that refine conjectures of
Aigner. The main tool is an explicit computation of the slopes at the corners
of the stable norm ball for the modular torus.
