Thin presentation of knots and lens spaces

Abstract

This paper concerns thin presentations of knots $K$ in closed $3$–manifolds $M^3$ which produce $S^3$ by Dehn surgery, for some slope $\gamma$. If $M$ does not have a lens space as a connected summand, we first 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 finally, if the core of the surgery $K_{\gamma }$ yields $S^3$ by $r$–Dehn surgery, then we prove the following inequality: $\left|r\right|\le 2g$, where $g$ is the genus of $K_{\gamma }$.