Bounds in simple hexagonal lattice and classification of 11-stick knots
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866909128980430848 |
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| author | Bao, Yueheng Benveniste, Ari Campisi, Marion Cazet, Nicholas Goh, Ansel Liu, Jiantong Sherman, Ethan |
| author_facet | Bao, Yueheng Benveniste, Ari Campisi, Marion Cazet, Nicholas Goh, Ansel Liu, Jiantong Sherman, Ethan |
| contents | The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Finally, we show that the only non-trivial 11-stick knots in the sh-lattice are the trefoil knot ($3_1$) and the figure-eight knot ($4_1$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_00687 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bounds in simple hexagonal lattice and classification of 11-stick knots Bao, Yueheng Benveniste, Ari Campisi, Marion Cazet, Nicholas Goh, Ansel Liu, Jiantong Sherman, Ethan Geometric Topology 57K10 The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Finally, we show that the only non-trivial 11-stick knots in the sh-lattice are the trefoil knot ($3_1$) and the figure-eight knot ($4_1$). |
| title | Bounds in simple hexagonal lattice and classification of 11-stick knots |
| topic | Geometric Topology 57K10 |
| url | https://arxiv.org/abs/2211.00687 |