On the number of components of twisted torus links
On the number of components of twisted torus links
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.
Adnan、Thiago de Paiva、Kyungbae Park
数学
Adnan,Thiago de Paiva,Kyungbae Park.On the number of components of twisted torus links[EB/OL].(2025-05-02)[2025-07-16].https://arxiv.org/abs/2505.01021.点此复制
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