#### Volume 11, issue 2 (2011)

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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 ${S}^{3}$ 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