Thin presentation of knots and lens spaces

This paper concerns thin presentations of knots $K$ in closed $3$ –manifolds $M^{3}$ which produce $S^{3}$ by Dehn surgery, for some slope $γ$ . 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 $K_{γ}$ yields $S^{3}$ by $r$ –Dehn surgery, then we prove the following inequality: $| r | \leq 2 g$ , where $g$ is the genus of $K_{γ}$ .

Keywords

Dehn surgery, lens space, thin presentation of knots, spines of 3–manifolds

Mathematical Subject Classification 2000

Primary: 57M25

Secondary: 57N10, 57M15

References

Forward citations

Publication

Received: 7 October 2002

Revised: 2 May 2003

Accepted: 9 June 2003

Published: 4 July 2003

Authors

A Deruelle
Université D’Aix-Marseille I C.M.I. 39 rue Joliot Curie Marseille Cedex 13 France

D Matignon
Université D’Aix-Marseille I C.M.I. 39 rue Joliot Curie Marseille Cedex 13 France

Volume 3, issue 2 (2003)

A Deruelle and D Matignon