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On a Remarkable Polyhedron Geometrizing the Figure Eight Knot Cone Manifolds

By Hugh Hilden


We define a one parameter family of polyhedra $P(t)$ that live in three dimensional spaces of constant curvature $C(t)$. Identifying faces in pairs in $P(t)$ via isometries gives rise to a cone manifold $M(t)$ (A cone manifold is much like an orbifold.). Topologically $M(t)$ is $S^3$ and it has a singular set that is the figure eight knot. As $t$ varies, curvature takes on every real value. At $t=-1$ a phenomenon which we call spontaneous surgery occurs and the topological type of $M(t)$ changes. We discuss the implications of this

Topics: 415, 57M50(MSC1991), 53C20(MSC1991)
Publisher: Graduate School of Mathematical Sciences, The University of Tokyo
Year: 1995
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Provided by: UT Repository
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