Multitriangulations on the half-cylinder

Saskia Solotko; Katherine Tung; Mengyuan Yang; Yuchong Zhang

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arXiv · 2506.16598

Multitriangulations on the half-cylinder

Saskia Solotko·Katherine Tung·Mengyuan Yang·Yuchong Zhang

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Abstract

We prove that the simplicial complex $\Delta _{\mathcal{C}_n,2}$ is pure and a weak pseudomanifold of dimension $2(n-1)$, where $\Delta _{\mathcal{C}_n,2}$ is the simplicial complex associated with $2$-triangulations on the half-cylinder with $n$ marked points. This result generalizes the work of Vincent Pilaud and Francisco Santos for polygons and resolves a conjecture of Mathias Lepoutre and Vincent Pilaud for $k=2$. To achieve this, we show that $2$-triangulations on the half-cylinder decompose as complexes of star polygons, and that $2$-triangulations on the half-cylinder are in bijection with $2$-triangulations on the $4n$-gon invariant under rotation by $\pi/2$ radians. Building on work by Vincent Pilaud and Christian Stump, we also introduce chevron pipe dreams, a new combinatorial model that more naturally captures the symmetries of $k$-triangulations.

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Saskia Solotko, Katherine Tung, Mengyuan Yang, Yuchong Zhang. 2025-06-19. Multitriangulations on the half-cylinder. https://arxiv.org/abs/2506.16598

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