Bridge number

Call an unbroken arc in this diagram a bridge if it includes at least one overcrossing.

[1] Bridge numbers were first studied in the 1950s by Horst Schubert.

[2] [3] The bridge number can equivalently be defined geometrically instead of topologically.

In bridge representation, a knot lies entirely in the plane apart for a finite number of bridges whose projections onto the plane are straight lines.

Equivalently, the bridge number is the minimal number of local maxima of the projection of the knot onto a vector, where we minimize over all projections and over all conformations of the knot.