The Bridge Index of a Knot

A bridge is a subarc of a knot diagram which contains only over-crossings. The bridge number of a knot diagram is minimum number of disjoint bridges which together include all overcrossings. This depends on the choice of diagram of a knot. For instance, the standard diagram of the trefoil knot has three bridges, but there is also a diagram with bridge number 2. The bridge index of a knot K is the minimum over all diagrams of K of the bridge numbers of those diagrams. This is denoted br(K). By convention br(unknot)= 1, and it is easily seen that any knot with bridge index equal to 1 is the unknot. Equivlently, one can define the bridge index of a knot to be the minimum over all knot diagrams of the number of local maxima of the knot, where the knot is viewed as a smooth closed curve in space.

A basic equality related to bridge index is the following:
Theorem (Schubert): If K1 and K2 are two knots then br(K1#K2)= br(K1) + br(K2) -1.

Knots with bridge index 2 are of particular interest since they have been completely classified (by Schubert) and include many of the knots under 11 crossings. There is no general method or algorithm for computing the bridge index of an arbitrary knot. This, in general, is a difficult problem.

Chad Musick has computed the bridge index for all 11-crossing knots. Those results are presented in the reference.

References

K. Murasugi, Knot Theory and Its Applications, Birkhauser Boston (1993)

C. Musick, Minimal Bridge Projections for 11-corssing prime knots. 2012 Arxiv Preprint

D. Rolfsen, Knots and Links, Publish or Perish, INC. Berkeley (1976)

H. Schubert, Uber eine nemerische Knoteninvariante, Math Zeit.61 (1954) 245-288