RT Journal Article
T1 On a remarkable polyhedron geometrizing the figure eight knot cone manifold
A1 Hilden, Hugh Michael
A1 Lozano Imízcoz, María Teresa
A1 Montesinos Amilibia, José María
AB The authors define a one-parameter family of polyhedra P(a), 0<a≤5−25√−−−−−−−√, in three-dimensional spaces of constant curvature −∞<k(a)≤1. Identifying faces of P(a) in pairs by isometries gives rise to cone manifolds M(a). For example, k=−1 when a=13−−√, and M gives the hyperbolic structure on the complement in S3 of the figure-eight knot K, k=0 when a=12−−√, and M gives the Euclidean structure on the orbifold which results from (3,0)-surgery on K, while k=1 when a=5−25√−−−−−−−√, and M gives the spherical structure on the orbifold which results from (2,0)-surgery on K. M(0) is a degenerate hyperbolic structure on the torus bundle B over S1 which results from (0,1)-surgery on K (let Σ⊂B denote the core circle of the surgery). The other M(a) interpolate between these, and after rescaling, as a increases, give hyperbolic structures on B, singular along Σ, with cone angles ranging from 2π to zero, then hyperbolic [resp. spherical] structures on S3, singular along K, with cone angles ranging from zero to 2π/3 [resp. 2π/3 to π]. Elementary formulas are derived for the volumes of the (rescaled) cone manifolds, the lengths of the (rescaled) singular sets, and the cone angles, as functions of a. Also, the phenomenon of "spontaneous surgery'' at a=13−−√ is linked to a combinatorial change in P(a).
PB Graduate School of Mathematical Sciences
SN 1340-5705
YR 1995
FD 1995
LK https://hdl.handle.net/20.500.14352/58635
UL https://hdl.handle.net/20.500.14352/58635
LA eng
NO Bonahon, F. and L. Siebenmann, The classification of Seifert fibred 3-orbifolds, London Math. Soc. LNSA95 (1985), 19–85.Bowditch, B. H., Maclachlan, C. and A. W. Reid, Arithmetic hyperbolic surface bundles, Preprint (1993).Burde, G. and Zieschang, H., Knots, Studies in Mathematics 5. de Gruyter, Berlin, New York, 1985.Coxeter, H. S. M., Non-Euclidean Geometry, University of Toronto Press, 1968.Dunbar, W. D., Geometricorbifolds, Revista Mat. Univ. Compl. Madrid 1 (1988), 67–99.Helling, H., Kim, A. C. and J. L. Mennicke, On Fibonacci groups, (to appear).Hilden, H. M., Lozano, M. T. and J. M. Montesinos-Amilibia, On the Borromean Orbifolds:Geometry and Arithmetic., TOPOLOGY’90. Ohio State University, Math. Research Inst. Pub. 1 (B. Apanasov, W. Neumann, A. Reid and L. Siebenmann, eds.), De Gruyter, 1992, pp. 133–167.Hilden, H. M., Lozano, M. T. and J. M. Montesinos-Amilibia, A Characterization of ArithmeticSubgroups of SL(2,R) and SL(2,C), Math. Nach. 159 (1992), 245–270.Hilden, H. M., Lozano, M. T. and J. M. Montesinos-Amilibia, The arithmeticity of the Figure Eigth knot orbifold., TOPOLOGY’90.Ohio State University, Math. Research Inst. Pub. 1 (B. Apanasov, W. Neumann, A. Reid and L. Siebenmann, eds.), De Gruyter, 1992, pp. 169-183.Hilden, H. M., Lozano, M. T. and J. M. Montesinos-Amilibia, The arithmeticity of certain torus bundle cone 3-manifolds, To appear.Hodgson, C., Degeneration and regeneration of geometricstructures on three-manifolds, Ph.D. Thesis, Princeton University (1986).Jorgensen, T., Compact 3-manifolds of constant negative curvature fibering over the circle, Annals of Math. 106 (1977), 61–72.Milnor, J., HyperbolicGeometry: The First 150 Years, Proc. of Symposia in Pure Mat. 39 (1983), 25–40.Milnor, J., The Schl¨affli differential equality, Preprint (1993).Milnor, J., Notes on hyperbolicvolume, in [T] (1978).Maclachlan, C. and Reid, A., Commesurability classes of arithmetic kleinian groups and their fuchsian subgroups, Math. Proc. Camb. Phil. Soc. 102 (1987), 251–257.Mednyck, A. D. and A. Ju. Vesnin, On compact and non-compact hyperbolicmanifolds with the same volume, Preprint (1992).Reid, A. W., Arithmetickleinian groups and their fuchsian subgroups, Ph.D. Thesis, Aberdeen (1987).Rolfsen, D., Knots and links, Publish or Perish, Inc., 1976.Scott, P., The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401–487.Selberg, A., On discontinuous groups in higher dimensional symmetric spaces, Int. Colloq. Function Theory. TATA Inst. Fundamental Res. Bombay (1960), 147–164.Thurston, W., The Geometry and Topology of 3-Manifolds, Notes 1976-1978. Princeton University Press (to appear).Vigneras, M. F., Arithmetique des Algebres de Quaternions, LNMA800, Springer-Verlag, 1980.Vinberg, E. B., Geometry II, Encyclopaedia of Mathematical Sciences. VolA29, Springer-Verlag, 1992.Wolfram, S., MATHEMATICA, A System for Doing Mathematics by Computer.
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RD 1 may 2024