Unexpected local minima in the width complexes for knots

Zupan, Alexander

doi:10.2140/agt.2011.11.1097

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Unexpected local minima in the width complexes for knots

Alexander Zupan

Algebraic & Geometric Topology 11 (2011) 1097–1105

DOI: 10.2140/agt.2011.11.1097

Abstract

In [Pacific J. Math. 239 (2009) 135–156], Schultens defines the width complex for a knot in order to understand the different positions a knot can occupy in $S^{3}$ and the isotopies between these positions. She poses several questions about these width complexes; in particular, she asks whether the width complex for a knot can have local minima that are not global minima. In this paper, we find an embedding of the unknot $0_{1}$ that is a local minimum but not a global minimum in the width complex for $0_{1}$ , resolving a question of Scharlemann. We use this embedding to exhibit for any knot $K$ infinitely many distinct local minima that are not global minima of the width complex for $K$ .

Keywords

width complex, thin position, unknot

Mathematical Subject Classification 2010

Primary: 57M25, 57M27

References

Publication

Received: 20 September 2010

Revised: 15 January 2011

Accepted: 16 January 2011

Published: 1 April 2011

Authors

Alexander Zupan
Department of Mathematics University of Iowa 14 MacLean Hall Iowa City IA 52242 USA
http://math.uiowa.edu/~azupan

Volume 11, issue 2 (2011)