#### 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.

Notes on file formats
A. Deruelle, D. Matignon

Universite D'Aix-Marseille I, C.M.I. 39, rue Joliot Curie

Marseille Cedex 13, France

Email: deruelle@cmi.univ-mrs.fr, matignon@cmi.univ-mrs.fr

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