Are torus links prime?

Question

It is known that if $p$ and $q$ are coprime integers, then the torus knot $T(p,q)$ is a prime knot. Are there any results of primeness of torus links? In particular, is it true that the torus links $T(2,n)$, where $n$ is an even integer, are prime links?

Answer 1

Due to geometrization, non-split links can be classified into three disjoint types: satellite links (i.e., links whose exterior has an incompressible non-boundary-parallel torus), hyperbolic links, and links with a Seifert-fibered exterior. Here's one starting point with some links.

Every connect sum is a satellite link by using a "swallow-follow" torus. Also, every torus link (assuming $p\neq 0$ or $q\neq 0$) is a non-split link with a Seifert-fibered exterior, so it is not a satellite, and hence not a connect sum.