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Unexpected local minima in the width complexes for knots

Alexander Zupan

Algebraic & Geometric Topology 11 (2011) 1097–1105
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 S3 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 01 that is a local minimum but not a global minimum in the width complex for 01, 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