An Algorithm to Find Ribbon Disks for Alternating Knots (Q6613274)

The authors devise an algorithm to find ribbon disks for alternating knots and implement it widely. One definition of a ribbon disk is a disk in \({\mathbb S}^3 \times I\), where projection onto the second factor yields a Morse function with respect to which the embedding of the disk has no minima. A band on a link is a square with opposite sides constituting two disjoint subintervals of the link. A band move consists of deleting the two subintervals of the link and replacing them with the other two sides of the square. Ribbon disks are characterized by their saddle points with respect to the Morse function and hence by band moves, because the band moves correspond to changes in level curves at saddle points.\N\NUsing one of the checkerboard surfaces of an alternating diagram, the authors check for bands of bounded complexity. The bands considered cross either two regions without twisting or one region with prescribed twisting. Surprisingly, checking for these low complexity bands suffices to find ribbon disks for instance for slice \(2\)-bridge knots. In addition, the algorithm finds slice ribbon disks for 662,903 prime alternating knots with 21 or fewer crossings. Related searches find several more slice ribbon disks for prime alternating knots with 21 or fewer crossings. In total, the authors resolve the question of whether a knot is ribbon for all but 3276 of the 1.2 billion prime alternating knots with 21 or fewer crossings.