Resumen:
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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)?lid?i, where V is the volume, and the sum is taken over all edges, li is the length of the ith edge and ?i 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 ? 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 ? (0????). There is an angle ?h such that the cone structure is hyperbolic when 0??
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