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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.05308 |
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| _version_ | 1866910402456059904 |
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| author | Baker, Kenneth L. Motegi, Kimihiko |
| author_facet | Baker, Kenneth L. Motegi, Kimihiko |
| contents | Twisting a given knot $K$ about an unknotted circle $c$ a full $n \in \mathbb{N}$ times, we obtain a "twist family" of knots $\{ K_n \}$. Work of Kouno-Motegi-Shibuya implies that for a non-trivial twist family the crossing numbers $\{c(K_n)\}$ of the knots in a twist family grows unboundedly. However potentially this growth is rather slow and may never become monotonic. Nevertheless, based upon the apparent diagrams of a twist family of knots, one expects the growth should eventually be linear. Indeed we conjecture that if $η$ is the geometric wrapping number of $K$ about $c$, then the crossing number of $K_n$ grows like $n η(η-1)$ as $n \to \infty$. To formulate this, we introduce the "stable crossing number" of a twist family of knots and establish the conjecture for (i) coherent twist families where the geometric wrapping and algebraic winding of $K$ about $c$ agree and (ii) twist families with wrapping number $2$ subject to an additional condition. Using the lower bound on a knot's crossing number in terms of its genus via Yamada's braiding algorithm, we bound the stable crossing number from below using the growth of the genera of knots in a twist family. (This also prompts a discussion of the "stable braid index".) As an application, we prove that highly twisted satellite knots in a twist family where the companion is twisted as well satisfy the Satellite Crossing Number Conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05308 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The stable crossing number of a twist family of knots and the satellite crossing number conjecture Baker, Kenneth L. Motegi, Kimihiko Geometric Topology 57K10, 57K14 Twisting a given knot $K$ about an unknotted circle $c$ a full $n \in \mathbb{N}$ times, we obtain a "twist family" of knots $\{ K_n \}$. Work of Kouno-Motegi-Shibuya implies that for a non-trivial twist family the crossing numbers $\{c(K_n)\}$ of the knots in a twist family grows unboundedly. However potentially this growth is rather slow and may never become monotonic. Nevertheless, based upon the apparent diagrams of a twist family of knots, one expects the growth should eventually be linear. Indeed we conjecture that if $η$ is the geometric wrapping number of $K$ about $c$, then the crossing number of $K_n$ grows like $n η(η-1)$ as $n \to \infty$. To formulate this, we introduce the "stable crossing number" of a twist family of knots and establish the conjecture for (i) coherent twist families where the geometric wrapping and algebraic winding of $K$ about $c$ agree and (ii) twist families with wrapping number $2$ subject to an additional condition. Using the lower bound on a knot's crossing number in terms of its genus via Yamada's braiding algorithm, we bound the stable crossing number from below using the growth of the genera of knots in a twist family. (This also prompts a discussion of the "stable braid index".) As an application, we prove that highly twisted satellite knots in a twist family where the companion is twisted as well satisfy the Satellite Crossing Number Conjecture. |
| title | The stable crossing number of a twist family of knots and the satellite crossing number conjecture |
| topic | Geometric Topology 57K10, 57K14 |
| url | https://arxiv.org/abs/2404.05308 |