On the number of components of twisted torus links
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908346116734976 |
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| author | Adnan de Paiva, Thiago Park, Kyungbae |
| author_facet | Adnan de Paiva, Thiago Park, Kyungbae |
| contents | Twisted torus links $T(p,q;r,s)$ generalize torus links by introducing $s$ additional twists on $r$ adjacent strands of the torus link $T(p,q)$. It is well known that the number of components of a torus link $T(p, q)$ is given by the greatest common divisor of $p$ and $q$. However, determining the number of components of twisted torus links is not as straightforward based solely on their parameters. In this work, we present a Euclidean algorithm-like procedure for computing the number of components of twisted torus links based on their parameters. As a result, we show that the number of components of a twisted torus link $T(p, q; r, s)$ is a multiple of $\gcd(p, q, r, s)$, and in particular, $T(p, q; r, s)$ is a knot only if $\gcd(p, q, r, s) = 1$. We also use our algorithm to prove several conjectures related to the number of components in twisted torus links. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01021 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the number of components of twisted torus links Adnan de Paiva, Thiago Park, Kyungbae Geometric Topology 57K10, 20B05 Twisted torus links $T(p,q;r,s)$ generalize torus links by introducing $s$ additional twists on $r$ adjacent strands of the torus link $T(p,q)$. It is well known that the number of components of a torus link $T(p, q)$ is given by the greatest common divisor of $p$ and $q$. However, determining the number of components of twisted torus links is not as straightforward based solely on their parameters. In this work, we present a Euclidean algorithm-like procedure for computing the number of components of twisted torus links based on their parameters. As a result, we show that the number of components of a twisted torus link $T(p, q; r, s)$ is a multiple of $\gcd(p, q, r, s)$, and in particular, $T(p, q; r, s)$ is a knot only if $\gcd(p, q, r, s) = 1$. We also use our algorithm to prove several conjectures related to the number of components in twisted torus links. |
| title | On the number of components of twisted torus links |
| topic | Geometric Topology 57K10, 20B05 |
| url | https://arxiv.org/abs/2505.01021 |