## On a Remarkable Polyhedron Geometrizing the Figure Eight Knot Cone Manifolds

J. Math. Sci. Univ. Tokyo
Vol. 2 (1995), No. 3, Page 501--561.

Hilden, Hugh ; Lozano, MarÃ­a Teresa ; Montesinos-Amilibia, JosÃ© MarÃ­a
On a Remarkable Polyhedron Geometrizing the Figure Eight Knot Cone Manifolds
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.