On a Remarkable Polyhedron Geometrizing the Figure Eight Knot Cone Manifolds
Vol. 2 (1995), No. 3, Page 501--561.
Hilden, Hugh ; Lozano, MarÃa Teresa ; Montesinos-Amilibia, José MarÃa
On a Remarkable Polyhedron Geometrizing the Figure Eight Knot Cone Manifolds
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Abstract:
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.
Mathematics Subject Classification (1991): 57M50, 53C20
Mathematical Reviews Number: MR1382519
Received: 1995-02-24
