Ribbon Disks for Alternating Knots

Citation

The paper, by Brendan Owens (University of Glasgow) and Frank Swenton (Middlebury College), presenting the details of this search and the mathematics from which this search arose, is currently available on arXiv. Please cite any use of these results via that paper:

http://arxiv.org/abs/2102.11778

The results of the search are provided below, and the data files provided can be opened with the KLO software; alternatively, users may test their own knots ad hoc via the control sequence ctrl-B as in this walkthrough.

Data files

The following files exhibit explicit ribbon disks for 231,996 algorithmically ribbon alternating knots up to 20 crossings, downloadable via the links below in zipped format. After unzipping, these .rb files can be opened in KLO in the Ribbon Disks dialog, which is accessed via the main menu Computation → Ribbon Disks; in the tree view presented, a knot can be double-clicked to open up the relevant diagram [sequence] as a new KLO document. The file below shows the "escape bands" for the 1730 identified escapees from the algorithm, as well as the Algorithm's bands and simplifications showing the result to be ribbon; the results of the first band moves are knot-equivalent to 2-component links that appear in the above search, thus exhibiting the original knot as ribbon (though a detour must be taken before the algorithm identifies the ribbon disk): For convenience, all of the above files are also available concatenated into a single zipped .rb file (note that escapees are still contained in a separate branch, not with the algorithmically ribbon knots):

Wanted!

All but 586 prime alternating knots of 20 crossings or fewer are either identified as ribbon knots via the Algorithm or escapee search or proven non-ribbon (via Fox-Milnor, HKL, or other obstructions); the remaining knots up to 20x are contained in this .rb file and consist of: It is expected that increasing proportions of the above knots starting at 18x are escapees requiring intermediate forms arising from algorithmically ribbon knots at 22x or higher, but the seven knots at 16x and 17x are expected to be obstructed in some way—we are offering a hefty US$10 bounty for results resolving the ribbon status of each of these seven knots!

For 21x Goeritz-bifactorisable prime alternating knots, we have records (i) of the specific obstructions for those known not to be slice and (ii) of the knots whose slice status remains unresolved; both are available by request.