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We computationally explore non-coherent band attachments between low crossingnumber knots, using grid diagrams. We significantly improve the currentH(2)-distance table. In particular, we find two new distance one pairs withfewer than seven crossings: one between $3_1\#3_1$ and $7_4m$, and a chirallycosmetic one for $7_3$. We further determine a total of 33 previously unknown$H(2)$-distance one pairs for knots with up to $8$ crossings. The appendix byKazuhiro Ichihara, In Dae Jong and Masakazu Teragaito contains a constructionexplaining the existence of chirally cosmetic bands for an infinite family ofknots, including $5_1,\, 7_3$ and $8_8$.

An exploration of low crossing and chiral cosmetic bands with grid diagrams