An algorithm to find ribbon disks for alternating knots

TL;DR

This paper presents an algorithm based on Donaldson's theorem to identify ribbon disks in alternating knots, successfully classifying sliceness for most small prime alternating knots through extensive computational analysis.

Contribution

The authors develop a novel algorithm leveraging Donaldson's diagonalisation theorem to find ribbon disks in alternating knots, significantly advancing computational knot theory.

Findings

Successfully finds ribbon disks for numerous classes of alternating knots.

Resolves sliceness for over 1.2 billion prime alternating knots with up to 21 crossings.

Identifies limitations and cases where the algorithm fails to find ribbon disks.

Abstract

We describe an algorithm to find ribbon disks for alternating knots, and the results of a computer implementation of this algorithm. The algorithm is underlain by a slice link obstruction coming from Donaldson's diagonalisation theorem. It successfully finds ribbon disks for slice two-bridge knots and for the connected sum of any alternating knot with its reverse mirror, as well as for 662,903 prime alternating knots of 21 or fewer crossings. We also identify some examples of ribbon alternating knots for which the algorithm fails to find ribbon disks, though a related search identifies all such examples known. Combining these searches with known obstructions, we resolve the sliceness of all but 3,276 of the over 1.2 billion prime alternating knots with 21 or fewer crossings.