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 628 identified escapees from the algorithm; 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):