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Let $(p,q)$ be a pair of coprime integers; the torus knot $K(p,q)$ is the unique (up to isotopy) curve on the boundary of a solid torus which is homologous to $p\cdot \lambda + q \cdot \mu$, where $\lambda, \mu$ are standard longitude and meridian of the solid torus. One can define torus knots even when $p$ and $q$ are not coprime, and one gets $n=\text{gcd}(p,q)$ copies of the torus knot $K(p/n, q/n)$; these links are called torus links. Let us suppose that $n=2$, so our torus link has two components. I would like to say that the linking number of these two components is non zero. An example of such a link is the Hopf link, which has linking number 1.Hopf Link

Indeed, I've found a formula (https://discover.wooster.edu/jbowen/files/2013/10/Total-linking-numbers-of-torus-links-and-Klein-links.pdf, Theorem 4.2) for the total linking number of torus links, which in the case of two components is the linking number of the components (which is what I need). The formula is

$$lk(T(m,n))=m \big(n- \frac{n}{gcd(m,n)}\big).$$

Ok, great. But for the Hopf link, which is a $T(2,2)$ torus link, the formula gives

$$lk(T(2,2))=2 \big(2-1)=2,$$

but the linking number of the Hopf link is 1! Am I missing something obvious?

P.S. If you know a simple way to say that the linking number of two-components torus links is nonzero, which does not involve any general formula, for me it's okay.

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I believe the paper you referenced has a typo in Theorem 4.2. It should say that the total linking number $\operatorname{lk}(T(m,n))$ of the torus link $T(m,n)$ is $$\operatorname{lk}(T(m,n)) = \frac{m}{2}\left(n - \frac{n}{\gcd(m,n)}\right).$$

The formula they give in Theorem 4.2 (without the extra $1/2$ factor) does not agree with their own computation in Figure 9. They compute $\operatorname{lk}(T(6,4)) = 6$ in Figure 9, but their own formula yields $$\operatorname{lk}(T(6,4)) = 6\left(4-\frac{4}{\gcd(6,4)}\right) = 12.$$

If you work through their proof, you'll see that they accurately count that the total number of crossings between different components of the $T(m,n)$ torus link is $m\left(n-\frac{n}{\gcd{m,n}}\right)$. Since all of those crossings are positive, it follows that the linking number is $$\operatorname{lk}(T(m,n)) = \frac{m}{2}\left(n - \frac{n}{\gcd(m,n)}\right).$$

Additional note: If all you are looking for is that the linking number is nonzero, then it suffices to know that torus links are positive, nonsplit links. The nonsplit part guarantees that there are crossings between different components, and the positive part tells you that all crossing count positively in the linking number sum.

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  • $\begingroup$ Perfect, great answer. Thank you very much:D $\endgroup$ Feb 3, 2021 at 9:51
  • $\begingroup$ I think the reference is wrong. For example for T(9,6) the total linking number should be 12, not 18. $\endgroup$
    – MathBug
    Dec 6, 2023 at 15:48

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