Piecewise circular curves and positivity

TL;DR

This paper introduces a moduli space of piecewise circular polygons in the Riemann sphere, relating it to Legendrian polygons and positive flag configurations, revealing topological and geometric structures.

Contribution

It establishes a connection between moduli spaces of piecewise circular polygons and positive flag configurations, showing the space is a topological ball and characterizing it geometrically.

Findings

The moduli space contains a component homeomorphic to the Fock-Goncharov space.

This component is a topological ball.

It is characterized as the space of simple piecewise circular curves with decreasing curvature.

Abstract

We introduce the moduli space of generic piecewise circular $n$-gons in the Riemann sphere and relate it to a moduli space of Legendrian polygons. We prove that when $n=2k$, this moduli space contains a connected component homeomorphic to the Fock-Goncharov space of $k$-tuples of positive flags for $\mathsf{PSp}(4,\mathbb{R})$ and hence is a topological ball. We characterize this component geometrically as the space of simple piecewise circular curves with decreasing curvature.