On volumes and Chern-Simons invariants of geometric 3-manifolds

Abstract

This paper presents a technique for computing Chern-Simons invariants of certain kinds of hyperbolic 3-manifolds, namely those which are obtained as n-fold branched covers of hyperbolic knots in S3. Let S(K,α) denote the hyperbolic cone manifold whose underlying topological space is S3 and whose singular locus is a geodesic isotopic to K with cone angle α. Such a manifold is obtained, for instance, by (generalized) hyperbolic Dehn surgery of type (rp,rq) on K with p,q relatively prime integers, and r real. If such a geometric object exists, it has cone angle 2rπ. Thurston's "orbifold theorem'' implies that if r≤1/2 and K satisfies certain topological conditions, then the cone manifold in question exists. Then the n-fold orbifold cover of S(K,2π/n) is the hyperbolic manifold Mn(K) obtained by n-fold cyclic covering of S3 branched over the knot K. The well-known "Schläfli formula'' says that the derivative of the volume of S(K,α) as a function of α is proportional to the length of the singular curve. The authors establish that the derivative of a generalization of the Chern-Simons invariant of S(K,α) as a function of α is proportional to the "jump'' of the singular curve—a generalization of the notion of torsion for a geodesic in a nonsingular hyperbolic manifold. This is proved by using the analyticity of the relevant quantities as functions on the PSL(2,C) representation variety of π1(S3−K), a result due to T. Yoshida [Invent. Math. 81 (1985), no. 3, 473–514;]. This together with the multiplicative nature of the Chern-Simons invariant for orbifolds allows one to calculate the Chern-Simons invariant of Mn(K).