## On Volumes and Chern-Simons Invariants of Geometric 3-Manifolds

J. Math. Sci. Univ. Tokyo
Vol. 3 (1996), No. 3, Page 723--744.

Hilden, Hugh M. ; Lozano, MarÃ­a Teresa ; Montesinos-Amilibia, JosÃ© MarÃ­a
On Volumes and Chern-Simons Invariants of Geometric 3-Manifolds
Let $M_n(K)$ be the hyperbolic 3-manifold obtained by $n$-cyclic covering of $S^3$ branched over a hyperbolic knot $K$. A method to compute the volume and the Chern-Simons invariant of $M_n(K)$ is given. The value of the volume of $M_n$ is $n$ times the value of the volume of the corresponding hyperbolic orbifold. This volume can be obtained by appying the SchlÃ¤ffli Formula for the volume to the cone-manifold family, $(K,Î± )$, with singularity $K$. The same approch is followed for the Chern-Simons invariant, after proving a "SchlÃ¤ffli Formula" for a generalized Chern-Simons function on the family of cone-manifold structures $(K,Î± )$.