Unexpected local minima in the width complexes for knots

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 $K$.