Let
denote the closed
orientable surface of genus
.
We construct exponentially many mapping class group orbits of pairs of simple closed curves
which fill
and intersect minimally, by showing that such orbits are in correspondence with the
solutions of a certain permutation equation in the symmetric group. Next, we
demonstrate that minimally intersecting filling pairs are combinatorially optimal, in
the sense that there are many simple closed curves intersecting the pair exactly
once. We conclude by initiating the study of a topological Morse function
over the moduli space of Riemann surfaces of genus
, which, given a
hyperbolic metric
,
outputs the length of the shortest minimally intersecting filling pair for the metric
.
We completely characterize the global minima of
and, using the exponentially many mapping class group orbits of minimally
intersecting filling pairs that we construct in the first portion of the paper,
we show that the number of such minima grows at least exponentially in
.
