Nerve complexes of circular arcs

TL;DR

This paper characterizes the homotopy types of nerve complexes of circular arcs, providing efficient algorithms for their computation and applications to topological bounds, graph coloring, and Vietoris--Rips complexes.

Contribution

It classifies the homotopy types of nerve complexes of circular arcs and offers algorithms to compute these types in optimal time, with applications to topology and graph theory.

Findings

Homotopy types are point, odd-dimensional sphere, or wedge sum of even-dimensional spheres.

Homotopy types can be computed in O(n log n) time.

Applications include bounds on roots of trigonometric polynomials, chromatic number of circular graphs, and Vietoris--Rips complexes.

Abstract

We show that the nerve complex of n arcs in the circle is homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same even dimension. Moreover this homotopy type can be computed in time O(n log n). For the particular case of the nerve complex of evenly-spaced arcs of the same length, we determine the dihedral group action on homology, and we relate the complex to a cyclic polytope with n vertices. We give three applications of our knowledge of the homotopy types of nerve complexes of circular arcs. First, we use the connection to cyclic polytopes to give a novel topological proof of a known upper bound on the distance between successive roots of a homogeneous trigonometric polynomial. Second, we show that the Lovasz bound on the chromatic number of a circular complete graph is either sharp or off by one. Third, we show that the Vietoris--Rips simplicial complex of n points in the circle is homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same even dimension, and furthermore this homotopy type can be computed in time O(n log n).