Nerve Complexes of Circular Arcs
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We show that the nerve and clique complexes of n arcs in the circle are 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 explicit homology bases 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 show that the Lovász bound on the chromatic number of a circular complete graph is either sharp or off by one. Second, 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. Third, we show that the Vietoris–Rips or ambient Čech 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 \(O(n\log n)\).
KeywordsNerve complex Čech complex Vietoris–Rips complex Circular arc Cyclic polytope
Mathematics Subject Classification05E45 52B15 68R05
We would like to thank Anton Dochtermann for encouraging us to consider the connection to the Lovász bound in Sect. 6, and we would like to thank Arnau Padrol and Yuliy Baryshnikov for helpful conversations about cyclic polytopes. We are grateful to the referees for suggestions regarding the paper, and in particular for bringing  to our attention. Research of MA was carried out while at the Max Planck Institut für Informatik, Saarbrücken, Germany. Research of HA was supported by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation. FF was supported by the German Science Foundation DFG via the Berlin Mathematical School.
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