Bridge Index

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 bridge number over all diagrams of K. 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.

Equivalently, 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: If K1 and K2 are two knots then br(K1#K2) = br(K1) + br(K2) - 1 [5].

Knots with bridge index 2 are of particular interest since they have been completely classified (by Schubert) and include many of the knots with fewer than 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 [3].

Ryan Blair, Alexandra Kjuchukova, Roman Velazquez, and Paul Villanueva have provided the bridge number for all 12-crossing knots [1].


[1] Blair, R., Kjuchukova, A., Velazquez, R., and Villanueva, P., "Wirtinger systems of generators of knot groups," Arxiv preprint.

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

[3] Musick, C., Minimal Bridge Projections for 11-crossing prime knots. Arxiv preprint.

[4] Rolfsen, D., Knots and Links, Publish or Perish, Inc. Berkeley (1976).

[5] Schubert, H., "Uber eine numerische Knoteninvariante," Math. Zeit. 61 (1954), 245-288.