Petal grid diagrams of torus knots
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866910657511686144 |
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| author | Lee, Eon-Kyung Lee, Sang-Jin |
| author_facet | Lee, Eon-Kyung Lee, Sang-Jin |
| contents | A petal diagram of a knot is a projection with a single multi-crossing such that there are no nested loops. The petal number $p(K)$ of a knot $K$ is the minimum number of loops among all petal diagrams of $K$.
Let $T_{n,s}$ denote the $(n,s)$-torus knot for relatively prime integers $2\le n<s$. Recently, Kim, No and Yoo proved that $p(T_{n,s})\le 2s-2\left\lfloor \frac sn\right\rfloor+1$ whenever $s\equiv \pm 1\bmod n$. They conjectured that the inequality holds without the assumption $s\equiv \pm 1\bmod n$. They also showed that $p(T_{n,s})=2s-1$ whenever $2\le n<s<2n$ and $n\equiv 1\bmod s-n$. Their proofs construct petal grid diagrams for those torus knots.
In this paper, we prove the conjecture that $p(T_{n,s})\le 2s-2\left\lfloor \frac sn\right\rfloor+1$ holds for any $2\le n<s$. We also show that $p(T_{n,s})=2s-1$ holds for any $2\le n<s<2n$. Our proofs construct petal grid diagrams for any torus knots. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_19486 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Petal grid diagrams of torus knots Lee, Eon-Kyung Lee, Sang-Jin Geometric Topology 57K10, 20F36 A petal diagram of a knot is a projection with a single multi-crossing such that there are no nested loops. The petal number $p(K)$ of a knot $K$ is the minimum number of loops among all petal diagrams of $K$. Let $T_{n,s}$ denote the $(n,s)$-torus knot for relatively prime integers $2\le n<s$. Recently, Kim, No and Yoo proved that $p(T_{n,s})\le 2s-2\left\lfloor \frac sn\right\rfloor+1$ whenever $s\equiv \pm 1\bmod n$. They conjectured that the inequality holds without the assumption $s\equiv \pm 1\bmod n$. They also showed that $p(T_{n,s})=2s-1$ whenever $2\le n<s<2n$ and $n\equiv 1\bmod s-n$. Their proofs construct petal grid diagrams for those torus knots. In this paper, we prove the conjecture that $p(T_{n,s})\le 2s-2\left\lfloor \frac sn\right\rfloor+1$ holds for any $2\le n<s$. We also show that $p(T_{n,s})=2s-1$ holds for any $2\le n<s<2n$. Our proofs construct petal grid diagrams for any torus knots. |
| title | Petal grid diagrams of torus knots |
| topic | Geometric Topology 57K10, 20F36 |
| url | https://arxiv.org/abs/2310.19486 |