Are the borromean rings a torus link?

Question

The $(3,3)$ torus link produces three unknots that are pairwise linked, which makes me suspect that the borromean rings ─ which are otherwise rather similar ─ cannot be produced through any $(p,q)$ torus link combination, and therefore that there is no deformation of the borromean rings that will make them lie on a torus. However, my knot theory isn't strong enough to make a harder case for this, and there's no mention of any connections in either wikipedia page.

So: can the borromean links be deformed so that they will lay on the surface of a torus? If so, how? If not, why not?

Answer 1

No, the Borromean rings are not a torus link, which is what we call links that can lie on a torus. To quote the wikipedia article on the Borromean rings:

The Borromean rings are a hyperbolic link: the complement of the Borromean rings in the 3-sphere admits a complete hyperbolic metric of finite volume.

Then we use the fact that every knot or link falls into one of three categories, this time using the wikipedia article on Hyperbolic links:

As a consequence of the work of William Thurston, it is known that every knot is precisely one of the following: hyperbolic, a torus knot, or a satellite knot. As a consequence, hyperbolic knots can be considered plentiful. A similar heuristic applies to hyperbolic links.

So, since the Borromean rings are hyperbolic, they cannot be a torus link. Hope this helps.