Thin presentation of knots and lens spaces

Abstract This paper concerns thin presentations of knots K in closed 3-manifolds M3 which produce S3 by Dehn surgery, for some slope γ. If M does not have a lens space as a connected summand, we rst prove that all such thin presentations, with respect to any spine of M have only local maxima. If M is a lens space and K has an essential thin presentation with respect to a given standard spine (of lens space M) with only local maxima, then we show that K is a 0-bridge or 1-bridge braid in M; furthermore, we prove the minimal intersection between K and such spines to be at least three, and nally, if the core of the surgery Kγ yields S3 by r-Dehn surgery, then we prove the following inequality: jrj 2g, where g is the genus of Kγ