Terminology for two trefoil knots linked together

Question

Is there a term describe the link formed by two trefoil knots (or any prime knots, I suppose) linked together like so?

Also, does that particular link correspond to a prime link?

Answer 1

The link formed by linking together the two trefoils (or any pair of nontrivial prime knots) is not prime. There is a sphere intersecting the link in exactly two points such that each side of the sphere is a non-trivial tangle:

Note that there are two ways to link two knots together:

These might be the same, but not in general. The case in which they are the same is when $K_1$ or $K_2$ is an invertible knot (an oriented knot $K$ is invertible if it is isotopic to $K$ with reversed orientation -- many small knots are invertible, like the trefoil).

As for what to call it, I might call it a plumbing of a split link with a Hopf link. Given the split link formed by $K_1$ and $K_2$,

then by plumbing in a Hopf link in the region between the two components, as portrayed by this diagram, the result is either of the two ways of linking the two knots together.

Another thing I might call it is a satellite operation, where $K_1$ and $K_2$ are patterns inserted into a Hopf link companion. This point of view comes from thinking of the linked knots as being the connect sum of $K_1$, $K_2$, and a Hopf link. It's true that connect sums are not well-defined for links, but it becomes well-defined if you say where the connect sums occur. The $K_1$ and $K_2$ knots are connect-summed into different components of the Hopf link.