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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.10892 |
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| _version_ | 1866913794910846976 |
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| author | Lamm, Christoph Eisermann, Michael |
| author_facet | Lamm, Christoph Eisermann, Michael |
| contents | The simultaneous crossing number is a new knot invariant which is defined for strongly invertible knots having diagrams with two orthogonal transvergent axes of strong inversions. Because the composition of the two inversions gives a cyclic period of order 2 with an axis orthogonal to the two axes of strong inversion, knot diagrams with this property have three characteristic orthogonal directions. We define the simultaneous crossing number, $\operatorname{sim}(K)$, as the minimum of the sum of the numbers of crossings of projections in the 3 directions, where the minimum is taken over all embeddings of $K$ satisfying the symmetry condition. Dividing the simultaneous crossing number by the usual crossing number, $\operatorname{cr}(K)$, of a knot gives a number $\ge 3$, because each of the 3 diagrams is a knot diagram of the knot in question. We show that $\liminf_{\operatorname{cr}(K) \to \infty} \operatorname{sim}(K)/\operatorname{cr}(K) \le 8$, when the minimum over all knots and the limit over increasing crossing numbers is considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10892 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiple points of view: The simultaneous crossing number for knots with doubly transvergent diagrams Lamm, Christoph Eisermann, Michael Geometric Topology The simultaneous crossing number is a new knot invariant which is defined for strongly invertible knots having diagrams with two orthogonal transvergent axes of strong inversions. Because the composition of the two inversions gives a cyclic period of order 2 with an axis orthogonal to the two axes of strong inversion, knot diagrams with this property have three characteristic orthogonal directions. We define the simultaneous crossing number, $\operatorname{sim}(K)$, as the minimum of the sum of the numbers of crossings of projections in the 3 directions, where the minimum is taken over all embeddings of $K$ satisfying the symmetry condition. Dividing the simultaneous crossing number by the usual crossing number, $\operatorname{cr}(K)$, of a knot gives a number $\ge 3$, because each of the 3 diagrams is a knot diagram of the knot in question. We show that $\liminf_{\operatorname{cr}(K) \to \infty} \operatorname{sim}(K)/\operatorname{cr}(K) \le 8$, when the minimum over all knots and the limit over increasing crossing numbers is considered. |
| title | Multiple points of view: The simultaneous crossing number for knots with doubly transvergent diagrams |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2504.10892 |