#### Volume 3, issue 2 (2003)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Thin presentation of knots and lens spaces

### A Deruelle and D Matignon

Algebraic & Geometric Topology 3 (2003) 677–707
 arXiv: math.GT/0402457
##### 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: $|r|\le 2g$, where $g$ is the genus of ${K}_{\gamma }$.

##### Keywords
Dehn surgery, lens space, thin presentation of knots, spines of 3–manifolds
##### Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57N10, 57M15