On manifolds with multiple lens space filings

TL;DR

This paper investigates the classification of 3-manifolds with multiple lens space fillings, examining the sharpness of the Cyclic Surgery Theorem through various specific manifold cases and their properties.

Contribution

It classifies non-hyperbolic manifolds with multiple lens space fillings and analyzes hyperbolic manifolds with three lens space fillings, clarifying the scope of the Cyclic Surgery Theorem.

Findings

Classified non-hyperbolic manifolds with more than one lens space filling.

Identified hyperbolic manifolds with three lens space fillings from the Minimally Twisted 5 Chain complement.

Showed that certain knots in $S^3$ and $S^1 imes S^2$ have no unexpected lens space surgeries.

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.