On a Remarkable Polyhedron Geometrizing the Figure Eight Knot Cone Manifolds

摘要

Wedefineaoneparameterfamilyofpolyhedra$P(t)$thatliveinthreedimensionalspacesofconstantcurvature$C(t)$.Identifyingfacesinpairsin$P(t)$viaisometriesgivesrisetoaconemanifold$M(t)$(Aconemanifoldismuchlikeanorbifold.).Topologically$M(t)$is$S^3$andithasasingularsetthatisthefigureeightknot.As$t$varies,curvaturetakesoneveryrealvalue.At$t=-1$aphenomenonwhichwecallspontaneoussurgeryoccursandthetopologicaltypeof$M(t)$changes.Wediscusstheimplicationsofthis.