On manifolds with multiple lens space filings

Abstract

An irreducible 3--manifold with torus boundary either is a Seifert fibered space or admits at most three lens space fillings according to the Cyclic Surgery Theorem. We examine the sharpness of this theorem by classifying the non-hyperbolic manifolds with more than one lens space filling, classifying the hyperbolic manifolds obtained by filling of the Minimally Twisted 5 Chain complement that have three lens space fillings, showing that the doubly primitive knots in $S^3$ and $S^1 \times S^2$ have no unexpected extra lens space surgery, and showing that the Figure Eight Knot Sister Manifold is the only non-Seifert fibered manifold with a properly embedded essential once-punctured torus and three lens space fillings.