An exploration of low crossing and chiral cosmetic bands with grid diagrams
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914254050820096 |
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| author | Barbensi, Agnese Celoria, Daniele Ktenidis, Christopher Ichihara, Kazuhiro Jong, In Dae Teragaito, Masakazu |
| author_facet | Barbensi, Agnese Celoria, Daniele Ktenidis, Christopher Ichihara, Kazuhiro Jong, In Dae Teragaito, Masakazu |
| contents | 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 $8_8$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_09749 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An exploration of low crossing and chiral cosmetic bands with grid diagrams Barbensi, Agnese Celoria, Daniele Ktenidis, Christopher Ichihara, Kazuhiro Jong, In Dae Teragaito, Masakazu Geometric Topology 57K10, 57-04 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 $8_8$. |
| title | An exploration of low crossing and chiral cosmetic bands with grid diagrams |
| topic | Geometric Topology 57K10, 57-04 |
| url | https://arxiv.org/abs/2504.09749 |