Flip-graphs of non-orientable filling surfaces
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918012286664704 |
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| author | Panda, Pallavi Parlier, Hugo Pournin, Lionel |
| author_facet | Panda, Pallavi Parlier, Hugo Pournin, Lionel |
| contents | Consider a surface $Σ$ with punctures that serve as marked points and at least one marked point on each boundary component. We build a filling surface $Σ_n$ by singling out one of the boundary components and denoting by $n$ the number of marked points it contains. We consider the triangulations of $Σ_n$ whose vertices are the marked points and the associated flip-graph $\mathcal{F}(Σ_n)$. Quotienting $\mathcal{F}(Σ_n)$ by the homeomorphisms of $Σ$ that fix the privileged boundary component results in a finite graph $\mathcal{MF}(Σ_n)$. Bounds on the diameter of $\mathcal{MF}(Σ_n)$ are available when $Σ$ is orientable and we provide corresponding bounds when $Σ$ is non-orientable. We show that the diameter of this graph grows at least like $5n/2$ and at most like $4n$ as $n$ goes to infinity. If $Σ$ is an unpunctured Möbius strip, $\mathcal{MF}(Σ_n)$ coincides with $\mathcal{F}(Σ_n)$ and we prove that the diameter of this graph grows exactly like $5n/2$ as $n$ goes to infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04074 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Flip-graphs of non-orientable filling surfaces Panda, Pallavi Parlier, Hugo Pournin, Lionel Geometric Topology Combinatorics Consider a surface $Σ$ with punctures that serve as marked points and at least one marked point on each boundary component. We build a filling surface $Σ_n$ by singling out one of the boundary components and denoting by $n$ the number of marked points it contains. We consider the triangulations of $Σ_n$ whose vertices are the marked points and the associated flip-graph $\mathcal{F}(Σ_n)$. Quotienting $\mathcal{F}(Σ_n)$ by the homeomorphisms of $Σ$ that fix the privileged boundary component results in a finite graph $\mathcal{MF}(Σ_n)$. Bounds on the diameter of $\mathcal{MF}(Σ_n)$ are available when $Σ$ is orientable and we provide corresponding bounds when $Σ$ is non-orientable. We show that the diameter of this graph grows at least like $5n/2$ and at most like $4n$ as $n$ goes to infinity. If $Σ$ is an unpunctured Möbius strip, $\mathcal{MF}(Σ_n)$ coincides with $\mathcal{F}(Σ_n)$ and we prove that the diameter of this graph grows exactly like $5n/2$ as $n$ goes to infinity. |
| title | Flip-graphs of non-orientable filling surfaces |
| topic | Geometric Topology Combinatorics |
| url | https://arxiv.org/abs/2505.04074 |