We study methods for computing the bridge number of a knot from a knot diagram. We prove equivalence between a geometric and a combinatorial definition of the bridge number of a knot diagram. 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.
I helped author the appendix to this article, “A note on Wirtinger number and flypes,” rather than the main article. My name therefore appears above the appendix instead of in the main author line.
