Volumes and Chern-Simons invariants of cyclic coverings over rational knots

Abstract

In this paper, the authors compute the volumes and Chern-Simons invariants for a class of hyperbolic 3-manifolds, namely, the n-fold branched covers of S3 along the 2-bridge knots p/q. The computation is based on the formula of Schläffli. In a 1-parameter family of polytopes in a space of constant curvature K, KdV=(1/2)∑lidαi, where V is the volume, and the sum is taken over all edges, li is the length of the ith edge and αi is its dihedral angle. Thus the volume of a 1-parameter family of cone-manifolds can be computed in terms of an initial volume and an integration involving length and cone angle of the singular curves. Similarly, the Chern-Simons invariant can be expressed in terms of an initial value and an integration involving the jump and the angle, based on earlier work of the authors. The 1-parameter family of cone-manifolds arises from the following. It is well-known that these 2-bridge knots have hyperbolic complements, which can be considered as hyperbolic cone-manifold structures on S3 with cone-angle 0 around the knot. It is also well-known that the 2-fold branched cover of S3 along p/q is the lens space Lp,q, which has spherical geometry, which induces a spherical cone-manifold structure on S3 with cone-angle π around the knot. These two structures are members of the family of cone-manifold structures on S3 having the 2-bridge knot p/q as a singular curve with angle α (0≤α≤π). There is an angle αh such that the cone structure is hyperbolic when 0≤α<αh, Euclidean when α=αh, and spherical when αh<α≤π. The authors choose the parameter to be x=2cosα, where α is the cone angle around the knot. They compute the functions jump, β(x), and length, δ(x), from the excellent component of the curve of representations of the knot group into SL(2,C). This allows them to compute the volumes and Chern-Simons invariants of the cone manifolds in terms of explicit integrals. The computation of the covering manifolds follows from the multiplicity of these invariants. Examples of numerical computations are shown at the end.