发表机构
Fordham University（福特汉姆大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了闭曲面上最小相交填充曲线系统的映射类群轨道数满足渐近公式$N_g \sim 16^g(2g)!/(64\pi\sqrt{2}g^3)$，并研究单曲线情形，改进上界，同时精确计算小亏格值并给出猜想。
AI 中文摘要
设$S_g$为亏格$g \geq 2$的闭可定向曲面，$\Gamma$为$S_g$上有限条闭曲线组成的集合，若其补集为若干圆盘的并，则称$\Gamma$填充曲面。在最小位置下，这样的$\Gamma$的双重点总数至少为$2g-1$；当等式成立时，我们称$\Gamma$为最小相交的。我们证明了最小相交填充曲线系统的映射类群轨道数$N_g$满足$N_g \sim 16^g (2g)!/(64\pi\sqrt{2}\\, g^3)$（当$g \to \infty$）。我们还研究了单曲线子问题，即$\Gamma$仅由一个连通分量组成。每条这样的曲线本身就是一个填充系统，因此相应的计数$N_g^{(1)}$满足$N_g^{(1)} \leq N_g$，这改进了先前已知的单条填充曲线的上界。最后，我们精确计算了小亏格下的$N_g$和$N_g^{(1)}$，并猜想其渐近增长率。
英文摘要
Let $S_g$ be the closed oriented surface of genus $g \geq 2$, 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 $N_g \sim 16^g (2g)!/(64π\sqrt{2}\, g^3)$ as $g \to \infty$. 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 $N_g^{(1)} \leq N_g$, 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.
Comments23 pages, 1 figure. Code and data are provided as ancillary files
