  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).
