Visibility graphs and deformations of associahedra

TL;DR

This paper generalizes the associahedron to arbitrary polygons using convex diagonalizations and visibility graphs, revealing new topological properties and deformation spaces related to polygon structures.

Contribution

It introduces a polytopal complex for any simple polygon based on convex diagonalizations and links it to visibility graphs and deformation spaces, extending associahedron theory.

Findings

Constructed a polytopal complex for arbitrary polygons.

Described topological properties and realizations of the complex.

Linked the complex to visibility graphs and deformation spaces.

Abstract

The associahedron is a convex polytope whose face poset is based on nonintersecting diagonals of a convex polygon. In this paper, given an arbitrary simple polygon P, we construct a polytopal complex analogous to the associahedron based on convex diagonalizations of P. We describe topological properties of this complex and provide realizations based on secondary polytopes. Moreover, using the visibility graph of P, a deformation space of polygons is created which encapsulates substructures of the associahedron.