Triangle-Free Triangulations

TL;DR

This paper investigates the structure of triangle-free triangulations of convex polygons, revealing a group action and lattice properties that help determine the flip graph's diameter.

Contribution

It establishes the transitive action of the affine Weyl group on colored triangulations and connects the flip graph to the weak order on this group.

Findings

Affine Weyl group acts transitively on triangulations

Flip graph related to a lower interval in the weak order

Computed the diameter of the flip graph

Abstract

The flip operation on colored inner-triangle-free triangulations of a convex polygon is studied. It is shown that the affine Weyl group $\widetilde{C}_n$ acts transitively on these triangulations by colored flips, and that the resulting colored flip graph is closely related to a lower interval in the weak order on $\widetilde{C}_n$. Lattice properties of this order are then applied to compute the diameter.