The Kakimizu complex for genus one hyperbolic knots in the 3-sphere
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866911030679961600 |
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| author | Valdez-Sánchez, Luis G. |
| author_facet | Valdez-Sánchez, Luis G. |
| contents | The Kakimizu complex $MS(K)$ for a knot $K\subset\mathbb{S}^3$ is the simplicial complex with vertices the isotopy classes of minimal genus Seifert surfaces in the exterior of $K$ and simplices any set of vertices with mutually disjoint representative surfaces. In this paper we determine the structure of the Kakimizu complex $MS(K)$ of genus one hyperbolic knots $K\subset\mathbb{S}^3$. In contrast with the case of hyperbolic knots of higher genus, it is known that the dimension $d$ of $MS(K)$ is universally bounded by $4$, and we show that $MS(K)$ consists of a single $d$-simplex for $d=0,4$ and otherwise of at most two $d$-simplices which intersect in a common $(d-1)$-face. For the cases $1\leq d\leq 3$ we also construct infinitely many examples of such knots where $MS(K)$ consists of two $d$-simplices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_09421 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Kakimizu complex for genus one hyperbolic knots in the 3-sphere Valdez-Sánchez, Luis G. Geometric Topology 57K10 The Kakimizu complex $MS(K)$ for a knot $K\subset\mathbb{S}^3$ is the simplicial complex with vertices the isotopy classes of minimal genus Seifert surfaces in the exterior of $K$ and simplices any set of vertices with mutually disjoint representative surfaces. In this paper we determine the structure of the Kakimizu complex $MS(K)$ of genus one hyperbolic knots $K\subset\mathbb{S}^3$. In contrast with the case of hyperbolic knots of higher genus, it is known that the dimension $d$ of $MS(K)$ is universally bounded by $4$, and we show that $MS(K)$ consists of a single $d$-simplex for $d=0,4$ and otherwise of at most two $d$-simplices which intersect in a common $(d-1)$-face. For the cases $1\leq d\leq 3$ we also construct infinitely many examples of such knots where $MS(K)$ consists of two $d$-simplices. |
| title | The Kakimizu complex for genus one hyperbolic knots in the 3-sphere |
| topic | Geometric Topology 57K10 |
| url | https://arxiv.org/abs/2311.09421 |