Algebraic and Geometric Topology 3 (2003), paper no. 23, pages 677-707.

Thin presentation of knots and lens spaces

A. Deruelle, D. Matignon

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| <= 2g, where g is the genus of K_gamma.

Keywords. Dehn surgery, lens space, thin presentation of knots, spines of 3-manifolds

AMS subject classification. Primary: 57M25. Secondary: 57N10, 57M15.

DOI: 10.2140/agt.2003.3.677

E-print: arXiv:math.GT/0402457

Submitted: 7 October 2002. (Revised: 2 May 2003.) Accepted: 9 June 2003. Published: 4 July 2003.

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A. Deruelle, D. Matignon
Universite D'Aix-Marseille I, C.M.I. 39, rue Joliot Curie
Marseille Cedex 13, France

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