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

The incompatibility of crossing number and bridge number for knot diagrams