Extrinsic systole of Seifert surfaces and distortion of knots
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917138407620608 |
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| author | Vasudevan, Sahana |
| author_facet | Vasudevan, Sahana |
| contents | In 1983, Gromov introduced the notion of distortion of a knot, and asked if there are knots with arbitrarily large distortion. In 2011, Pardon proved that the distortion of $T_{p,q}$ is at least $\min\{p,q\}$ up to a constant factor. We prove that the distortion of $T_{p, p+1}\# K$ is at least $p$ up to a constant, independent of $K$. We also prove that any embedding of a minimal genus Seifert surface for $T_{p,p+1}\# K$ in $\mathbb{R}^3$ has small extrinsic systole, in the sense that it contains a non-contractible loop with small $\mathbb{R}^3$-diameter relative to the length of the knot. These results are related to combinatorial properties of the monodromy map associated to torus knots. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_01095 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extrinsic systole of Seifert surfaces and distortion of knots Vasudevan, Sahana Geometric Topology Differential Geometry Metric Geometry In 1983, Gromov introduced the notion of distortion of a knot, and asked if there are knots with arbitrarily large distortion. In 2011, Pardon proved that the distortion of $T_{p,q}$ is at least $\min\{p,q\}$ up to a constant factor. We prove that the distortion of $T_{p, p+1}\# K$ is at least $p$ up to a constant, independent of $K$. We also prove that any embedding of a minimal genus Seifert surface for $T_{p,p+1}\# K$ in $\mathbb{R}^3$ has small extrinsic systole, in the sense that it contains a non-contractible loop with small $\mathbb{R}^3$-diameter relative to the length of the knot. These results are related to combinatorial properties of the monodromy map associated to torus knots. |
| title | Extrinsic systole of Seifert surfaces and distortion of knots |
| topic | Geometric Topology Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2510.01095 |