The incompatibility of crossing number and bridge number for knot diagrams

TL;DR

This paper investigates various methods of computing the bridge number of knot diagrams, revealing that minimal crossing diagrams can significantly differ from the knot's true bridge number and that this discrepancy can grow arbitrarily large.

Contribution

It introduces multiple natural definitions of diagrammatic bridge number and demonstrates their divergence from the actual bridge number, including constructions with unbounded differences.

Findings

Crossing number minimizing diagrams can fail to minimize bridge number.

The difference between diagrammatic and actual bridge number can grow infinitely.

Certain minimal crossing diagrams do not reflect the true bridge number of the knot.

Abstract

We define and compare several natural ways to compute the bridge number of a knot diagram. We study bridge numbers of crossing number minimizing diagrams, as well as the behavior of diagrammatic bridge numbers under the connected sum operation. For each notion of diagrammatic bridge number considered, we find crossing number minimizing knot diagrams which fail to minimize bridge number. Furthermore, we construct a family of minimal crossing diagrams for which the difference between diagrammatic bridge number and the actual bridge number of the knot grows to infinity.