Circular thin position for knots in S3

Manjarrez-Gutiérrez, Fabiola

doi:10.2140/agt.2009.9.429

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Circular thin position for knots in $S^3$

Fabiola Manjarrez-Gutiérrez

Algebraic & Geometric Topology 9 (2009) 429–454

DOI: 10.2140/agt.2009.9.429

Abstract

A regular circle-valued Morse function on the knot complement $C_{K} = S^{3} ∖ K$ is a function $f : C_{K} \to S^{1}$ which separates critical points and which behaves nicely in a neighborhood of the knot. Such a function induces a handle decomposition on the knot exterior $E (K) = S^{3} ∖ N (K)$ , with the property that every regular level surface contains a Seifert surface for the knot. We rearrange the handles in such a way that the regular surfaces are as “simple" as possible. To make this precise the concept of circular width for $E (K)$ is introduced. When $E (K)$ is endowed with a handle decomposition which realizes the circular width we will say that the knot $K$ is in circular thin position. We use this to show that many knots have more than one nonisotopic incompressible Seifert surface. We also analyze the behavior of the circular width under some knot operations.

Keywords

thin position, knots, Seifert surfaces, circle-valued Morse functions

Mathematical Subject Classification 2000

Primary: 57M25

References

Publication

Received: 20 October 2008

Revised: 1 December 2008

Accepted: 16 January 2009

Published: 6 March 2009

Authors

Fabiola Manjarrez-Gutiérrez
Department of Mathematics University of California Davis, CA, 95616 USA
http://www.math.ucdavis.edu/~fabiola

Volume 9, issue 1 (2009)