Multitriangulations on the half-cylinder

TL;DR

This paper proves that the complex of 2-triangulations on the half-cylinder is a pure weak pseudomanifold, generalizing polygon results and introducing chevron pipe dreams to better understand symmetries.

Contribution

It extends the theory of triangulations to half-cylinders, proves a conjecture for k=2, and introduces chevron pipe dreams as a new combinatorial model.

Findings

The complex is pure and a weak pseudomanifold of dimension 2(n-1).

2-triangulations on the half-cylinder correspond to those on a 4n-gon with rotational symmetry.

Introduces chevron pipe dreams to capture symmetries of k-triangulations.

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.