Multitriangulations on the half-cylinder
Saskia Solotko, Katherine Tung, Mengyuan Yang, Yuchong Zhang
TL;DR
This work extends the theory of multitriangulations from polygons to the half-cylinder $\mathcal{C}_n$, defining weak $k$-triangulations via projections of periodic $(k+1)$-crossing-free sets on the universal cover. It proves a Star Decomposition Theorem for $k=2$, yielding purity and the weak pseudomanifold property for $\Delta_{\mathcal{C}_n,2}$ and establishing regularity of the flip graph, thereby generalizing polygonal results to a surface with boundary. A canonical bijection $\phi$ relates $2$-triangulations on $\mathcal{C}_n$ to $n$-periodic $2$-triangulations on the $4n$-gon, enabling transfer of combinatorial structure and counts; this is complemented by the introduction of chevron pipe dreams to capture central symmetries and connect to subword complexes. The findings suggest broader generalizations to arbitrary $k$ and other surfaces, and open a pathway to representing finite multitriangulation complexes as (piecewise) linear spheres via symmetric, combinatorial models.
Abstract
We prove that the simplicial complex $Δ_{\mathcal{C}_n,2}$ is pure and a weak pseudomanifold of dimension $2(n-1)$, where $Δ_{\mathcal{C}_n,2}$ 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 $π/2$ 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.