Spinal Knots in Lens Spaces

Abstract

A knot in a lens is said to be spinal if it can be isotoped on a standard spine (e.g. in RP 3 , spinal knots bound a Möbius band). We prove that a Dehn surgery on a non-spinal knot in a lens space cannot produce S 3 . With a view to study the Dehn surgeries that produce lens spaces, the main part is devoted to finding an obstruction for a standard spine to be minimal. We consider the intersection graphs coming from a standard spine and an arbitrary surface. This obstruction is given by the existence of a generalized Scharlemann cycle.