Unexpected local minima in the width complexes for knots

TL;DR

This paper demonstrates the existence of local minima in the width complexes of knots, including the unknot, showing that such minima can be distinct from the global minimum, thus answering a question posed by Schultens.

Contribution

It constructs explicit examples of local minima in the width complexes for knots, including infinitely many for any given knot, revealing new complexity in knot position classifications.

Findings

Existence of local minima in the width complex of the unknot.

Construction of infinitely many local minima for any knot.

Local minima are not necessarily global minima in width complexes.

Abstract

In "Width complexes for knots and 3-manifolds," Jennifer Schultens defines the width complex for a knot in order to understand the different positions a knot can occupy in the 3-sphere 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 its width complex. 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.