Hosting and Friendship of Knots on Minimal Genus Seifert Surfaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911569480253440 |
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| author | Ozawa, Makoto |
| author_facet | Ozawa, Makoto |
| contents | For a knot $K\subset S^3$, let $S(K)$ denote the set of knot types represented by simple closed curves on a minimal genus Seifert surface of $K$. We study the directed relation $K\to J$ defined by $J\in S(K)$, which we call the \emph{hosting relation}, and call its symmetric part friendship. This gives a new framework for describing how knots appear on minimal genus Seifert surfaces of other knots.
A classical result of Lyon implies that the family of torus knots is a universal host family: every non-trivial knot is hosted by some torus knot. In contrast, a central result of this paper is that no knot is a universal host: for every knot $K$, there exists a knot $J$ such that \[ J\notin S(K). \] Thus universal hosting occurs at the level of families, but never at the level of a single knot.
We also study explicit examples of hosting and friendship. In particular, we describe the hosting set of the trefoil in terms of primitive slope classes on its once-punctured torus fiber, and use this description to obtain concrete friendship and non-friendship phenomena. For example, we show that $3_1$ and $8_{19}$ are friends, whereas $3_1$ and $4_1$ are not.
These results provide a framework for studying universal host phenomena, hosting, and friendship among knots on minimal genus Seifert surfaces, and suggest further connections with graph-theoretic, rigidity, and categorical aspects of knot theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04657 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hosting and Friendship of Knots on Minimal Genus Seifert Surfaces Ozawa, Makoto Geometric Topology For a knot $K\subset S^3$, let $S(K)$ denote the set of knot types represented by simple closed curves on a minimal genus Seifert surface of $K$. We study the directed relation $K\to J$ defined by $J\in S(K)$, which we call the \emph{hosting relation}, and call its symmetric part friendship. This gives a new framework for describing how knots appear on minimal genus Seifert surfaces of other knots. A classical result of Lyon implies that the family of torus knots is a universal host family: every non-trivial knot is hosted by some torus knot. In contrast, a central result of this paper is that no knot is a universal host: for every knot $K$, there exists a knot $J$ such that \[ J\notin S(K). \] Thus universal hosting occurs at the level of families, but never at the level of a single knot. We also study explicit examples of hosting and friendship. In particular, we describe the hosting set of the trefoil in terms of primitive slope classes on its once-punctured torus fiber, and use this description to obtain concrete friendship and non-friendship phenomena. For example, we show that $3_1$ and $8_{19}$ are friends, whereas $3_1$ and $4_1$ are not. These results provide a framework for studying universal host phenomena, hosting, and friendship among knots on minimal genus Seifert surfaces, and suggest further connections with graph-theoretic, rigidity, and categorical aspects of knot theory. |
| title | Hosting and Friendship of Knots on Minimal Genus Seifert Surfaces |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2604.04657 |