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

Agnese Barbensi; Daniele Celoria; Christopher Ktenidis; Kazuhiro Ichihara; In Dae Jong; Masakazu Teragaito

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arXiv · 2504.09749

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

Agnese Barbensi·Daniele Celoria·Christopher Ktenidis·Kazuhiro Ichihara·In Dae Jong·Masakazu Teragaito

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Abstract

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 fewer than seven crossings: one between $3_1\#3_1$ and $7_4m$, and a chirally cosmetic 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 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 $5_1,\, 7_3$ and $8_8$.

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Agnese Barbensi, Daniele Celoria, Christopher Ktenidis, Kazuhiro Ichihara, In Dae Jong, Masakazu Teragaito. 2025-04-13. An exploration of low crossing and chiral cosmetic bands with grid diagrams. https://arxiv.org/abs/2504.09749

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