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 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 that is a local minimum but not a global minimum in the width complex for , resolving a question of Scharlemann. We use this embedding to exhibit for any knot infinitely many distinct local minima that are not global minima of the width complex for .
"Unexpected local minima in the width complexes for knots." Algebr. Geom. Topol. 11 (2) 1097 - 1105, 2011. https://doi.org/10.2140/agt.2011.11.1097