A Linear Bound on the Diameter of the Kakimizu Complex for Hyperbolic Knots
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911444125089792 |
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| author | Chen, Xiao Shen, Wujie |
| author_facet | Chen, Xiao Shen, Wujie |
| contents | This paper focuses on the Kakimizu complex of a hyperbolic knot $K$. We define a complex $IS_\ell(K)$ to study incompressible Seifert surfaces of genus at most $\ell$, and prove that it is connected and that its diameter admits a linear upper bound in terms of $\ell$. As a corollary, we show that the diameter of the Kakimizu complex $MS(K)$ of a hyperbolic knot grows linearly with the genus $g$, confirming a conjecture of Sakuma--Shackleton. More precisely, it is bounded above by $6g-4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_03353 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Linear Bound on the Diameter of the Kakimizu Complex for Hyperbolic Knots Chen, Xiao Shen, Wujie Geometric Topology This paper focuses on the Kakimizu complex of a hyperbolic knot $K$. We define a complex $IS_\ell(K)$ to study incompressible Seifert surfaces of genus at most $\ell$, and prove that it is connected and that its diameter admits a linear upper bound in terms of $\ell$. As a corollary, we show that the diameter of the Kakimizu complex $MS(K)$ of a hyperbolic knot grows linearly with the genus $g$, confirming a conjecture of Sakuma--Shackleton. More precisely, it is bounded above by $6g-4$. |
| title | A Linear Bound on the Diameter of the Kakimizu Complex for Hyperbolic Knots |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2508.03353 |