Asymptotic enumeration of minimally intersecting filling curve systems on closed surfaces
Abstract
Let Sg be the closed oriented surface of genus g ≥ 2, and let Γ be a finite collection of closed curves on Sg that fills, in the sense that its complement is a union of disks. The total number of double points of such a Γ in minimal position is at least 2g−1; we call Γ minimally intersecting when equality holds. We prove that the number Ng of mapping class group orbits of minimally intersecting filling curve systems satisfies Ng ∼ 16g (2g)!/(64π√2 g3) as g → ∞. We also study the single-curve subproblem, in which Γ consists of a single component. Every such curve is in particular a filling system, so the corresponding count Ng(1) satisfies Ng(1) ≤ Ng, which improves the upper bound previously known for single filling curves. Finally, we compute Ng and Ng(1) exactly for small genus and conjecture an asymptotic growth rate.
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BibTeX
@misc{mondal2026asymptotic,
title = {{Asymptotic enumeration of minimally intersecting filling curve systems on closed surfaces}},
author = {Sayantika Mondal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09102},
eprint = {2610.09102},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09102},
}
RIS
TY - UNPB
TI - Asymptotic enumeration of minimally intersecting filling curve systems on closed surfaces
AU - Sayantika Mondal
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.09102
UR - https://arxiv.org/abs/2610.09102
AB - Let S_g be the closed oriented surface of genus g ≥ 2, and let Γ 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 Γ in minimal position is at least 2g−1; we call Γ 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 ∼ 16^g (2g)!/(64π√2 g^3) as g → ∞. We also study the single-curve subproblem, in which Γ 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) ≤ 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.
ER -