Cross-ratio degrees and triangulations

TL;DR

This paper provides a closed-form formula for a class of cross-ratio degrees associated with triangulations of polygons, linking combinatorics, geometry, and positive geometry in the context of configuration spaces.

Contribution

It introduces a simple closed formula for cross-ratio degrees indexed by polygon triangulations, enhancing understanding of their structure and connections to geometric spaces.

Findings

Derived a closed formula for specific cross-ratio degrees

Connected cross-ratio degrees to the geometry of M_{0,n}

Linked the degrees to positive geometry

Abstract

The cross-ratio degree problem counts configurations of n points on P^1 with n-3 prescribed cross-ratios. Cross-ratio degrees arise in many corners of combinatorics and geometry, but their structure is not well-understood in general. Interestingly, examining various special cases of the problem can yield combinatorial structures that are both diverse and rich. In this paper we prove a simple closed formula for a class of cross-ratio degrees indexed by triangulations of an n-gon; these degrees are connected to the geometry of the real locus of M_{0,n}, and to positive geometry.