Abstract
In the present article, we obtain some explicit integral formulas for the generalized Chern-Simons function I(W(α,β)) for Whitehead link cone-manifolds in the hyperbolic and spherical cases. We also give the Chern-Simons invariant for the Whitehead link orbifolds. We find a formula for the Chern-Simons invariant of n-fold coverings of the three-sphere branched over the Whitehead link.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Coulson D., Goodman O. A., Hodgson C. D., and Neumann W. D., “Computing arithmetic invariants of 3-manifolds,” Experiment. Math. 9(1), 127–152 (2000).
Culler M., “Lifting representations to covering groups,” Adv. Math. 59(1), 64–79 (1986).
Fenchel W., Elementary Geometry in Hyperbolic Space, Vol. 11 (Walter de Gruyter & Co., Berlin, 1989).
Hilden H. M., Lozano M. T., and Montesinos-Amilibia J. M., “On a remarkable polyhedron geometrizing the figure eight knot cone manifolds,” J. Math. Sci. Univ. Tokyo 2(3), 501–561 (1995).
Hilden H. M., Lozano M. T., and Montesinos-Amilibia J. M., “On the arithmetic 2-bridge knots and link orbifolds and a new knot invariant,” J. Knot Theory Ramifications 4(1), 81–114 (1995).
Hilden H. M., Lozano M. T., and Montesinos-Amilibia J. M., “On volumes and Chern-Simons invariants of geometric 3-manifolds,” J. Math. Sci. Univ. Tokyo 3(3), 723–744 (1996).
Hilden H. M., Lozano M. T., and Montesinos-Amilibia J. M., “Volumes and Chern-Simons invariants of cyclic covering over rational knots” in Topology and Teichmüller Spaces. Proc. of the 37th Taniguchi Symposium held in Finland, July 1995 (Singapore: World Scientific, 1996), pp. 31–35.
Hodgson C. D., Schläfli Revisited: Variation of Volume in Constant Curvature Spaces, Manuscript (University of Melbourne, Melbourne, 1991).
Kojima S. “Deformations of hyperbolic 3-cone-manifolds,” J. Differential Geom. 49(3), 469–516 (1998).
Mednykh A. “On the remarkable properties of the hyperbolic Whitehead link cone-manifold” in Knots in Hellas’98, Vol. 24 of Proc. of the Int. Conf. on Knot Theory and Its Ramifications (World Scientific, Singapore, 2000), pp. 290–305.
Meyerhoff R., “Density of the Chern-Simons invariant for hyperbolic 3-manifolds” in Low Dimensional Topology and Kleinian Groups, Vol. 112 of London Math. Soc. Lecture Note Ser. (Cambridge Univ. Press, Cambridge, 1986), pp. 217–239.
Mostow G. D., “Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms,” Inst. Hautes Études Sci. Publ. Math. 34, 53–104 (1968).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © N.V. Abrosimov, 2007, published in Matematicheskie Trudy, 2007, Vol. 10, No. 1, pp. 3–15.
From now on, by “geometric” we mean a structure that admits a metric of constant (positive, negative, or zero) curvature.
About this article
Cite this article
Abrosimov, N.V. The Chern-Simons invariants of cone-manifolds with Whitehead link singular set. Sib. Adv. Math. 18, 77–85 (2008). https://doi.org/10.3103/S1055134408020016
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1055134408020016