An Exploration of Low Crossing and Chiral Cosmetic Bands with Grid Diagrams

We computationally explore non-coherent band attachments between low crossing number knots, using grid diagrams. We significantly improve the current H(2)-distance table. In particular, we find two new distance one pairs with at most seven crossings: one between 31#31 and 74m, and a chirally cosmetic one for 73. We further determine a total of 33 previously unknown H(2)-distance one pairs for knots with up to 8 crossings. The appendix by Kazuhiro Ichihara, In Dae Jong and Masakazu Teragaito contains a construction explaining the existence of chirally cosmetic bands for an infinite family of knots, including 51,73, and 88.