Polyhedral Deformations of Cone Manifolds

Two single parameter families of polyhedra $P(\psi )$ are constructed in three dimensional spaces of constant curvature $C(\psi )$. Identification of the faces of the polyhedra via isometries results in cone manifolds $M(\psi )$ which are topologically $S^1{\text{timesS}}^2$, $S^3$ or singular $S^2$. The singular set of $M(\psi )$ can have self intersections for some values of $\psi$ and can also be the Whitehead link or form other configurations. Curvature varies continuously with $\psi$. At $\psi =0$ spontaneous surgery occurs and the topological type of $M(\psi )$ changes. This phenomenon is described.