On the tree-width of knot diagrams

TL;DR

This paper links the tree-width of knot diagrams to the topological complexity of the knots, showing that small tree-decompositions imply simple surfaces, and provides examples of knots with inherently high diagram complexity.

Contribution

It establishes a connection between tree-width of knot diagrams and the existence of simple topological surfaces, answering longstanding questions about diagram complexity.

Findings

Small tree-decomposition induces small sphere-decomposition

Existence of knots with high tree-width in all diagrams

First examples of knots with inherently high diagram complexity

Abstract

We show that a small tree-decomposition of a knot diagram induces a small sphere-decomposition of the corresponding knot. This, in turn, implies that the knot admits a small essential planar meridional surface or a small bridge sphere. We use this to give the first examples of knots where any diagram has high tree-width. This answers a question of Burton and of Makowsky and Mari\~no.