Arcs on Punctured Disks Intersecting at Most Twice with Endpoints on the Boundary

Assaf Bar-Natan

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arXiv · 1708.06402

Arcs on Punctured Disks Intersecting at Most Twice with Endpoints on the Boundary

Assaf Bar-Natan

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Abstract

Let $D_n$ be the $n$-punctured disk. We prove that a family of essential simple arcs starting and ending at the boundary and pairwise intersecting at most twice is of size at most $\binom{n+1}{3}$. On the way, we also show that any nontrivial square complex homeomorphic to a disk whose hyperplanes are simple arcs intersecting at most twice must have a corner or a spur.

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Assaf Bar-Natan. 2017-08-21. Arcs on Punctured Disks Intersecting at Most Twice with Endpoints on the Boundary. https://arxiv.org/abs/1708.06402

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