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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 M3 which produce S3 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 Kγ yields S3 by r–Dehn surgery, then we prove the following inequality: |r| 2g, where g is the genus of Kγ.

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