Asymptotic enumeration of minimally intersecting filling curve systems on closed surfaces

Abstract

Let $S_g$ be the closed oriented surface of genus $g\ge 2$, and let $\Gamma$ be a finite collection of closed curves on $S_g$ that fills, in the sense that its complement is a union of disks. The total number of double points of such a $\Gamma$ in minimal position is at least $2g-1$; we call $\Gamma$ minimally intersecting when equality holds. We prove that the number $N_g$ of mapping class group orbits of minimally intersecting filling curve systems satisfies $N_g\sim 16^g(2g)!/(64\pi \sqrt{2}{g}^3)$ as $g\to \infty$. We also study the single-curve subproblem, in which $\Gamma$ consists of a single component. Every such curve is in particular a filling system, so the corresponding count $N_g^{(1)}$ satisfies $N_g^{(1)}\le N_g$, which improves the upper bound previously known for single filling curves. Finally, we compute $N_g$ and $N_g^{(1)}$ exactly for small genus and conjecture an asymptotic growth rate.

arXiv comment23 pages, 1 figure. Code and data are provided as ancillary files