# Transformed flips in triangulations and matchings

## TL;DR

This paper explores the relationship between edge flips in triangulations and perfect matchings of convex point sets, providing a detailed algebraic and combinatorial analysis of these transformations.

## Contribution

It establishes a precise correspondence between edge flips in triangulations and matchings, and interprets the flip graph algebraically via the Temperley-Lieb algebra.

## Key findings

- Explicit bijection between triangulation flips and matching flips
- Algebraic interpretation of flip graph using Temperley-Lieb algebra
- Characterization of local changes in combinatorial structures

## Abstract

Plane perfect matchings of $2n$ points in convex position are in bijection with triangulations of convex polygons of size $n+2$. Edge flips are a classic operation to perform local changes both structures have in common. In this work, we use the explicit bijection from Aichholzer et al. (2018) to determine the effect of an edge flip on the one side of the bijection to the other side, that is, we show how the two different types of edge flips are related. Moreover, we give an algebraic interpretation of the flip graph of triangulations in terms of elements of the corresponding Temperley-Lieb algebra.