We define a solvable link as a link that can be built up iterating cabling and connected sums from the unknot in $S^3$. I recall the definition of cabling: a $(p,q)$-cabling of a knot $K$, where $p,q$ are coprime integers, consists in picking a natural number $d\geq 1$ and doing one of the following operations:
we replace $K$ with a satellite whose pattern is the torus knot (or link) $d\cdot K(p,q)$ in a standard solid torus and whose companion is the knot $K$ (we call it $d \cdot \widetilde{K}(p,q))$. Here, we denote with $d\cdot K(p,q)$ the union of $d$ parallel copies of the torus knot $K(p,q)$, which is the torus link $K(dp, dq)$. For example, if $K$ is the unknot, the result of this operation is replacing the unknot with a torus knot or a torus link;
we replace $K$ with the link $L= K \cup d\cdot \widetilde{K}(p,q)$.
The connected sum of knots is defined in the common way, whether as connected sum of links we mean that we choose a component of each link we are summing, an interval of each one of these components and then we perform the connected sum of knots; I think this operation is not well defined, however we allow it anyway.
So, let's see what we got when performing these operations on the unknot:
the unknot (obviously), torus knots and connected sums of torus knots;
iterated torus knots;
torus links, ecc...
Now we come to my point: $(p,q)$ are coprime integers, but do we allow one of them to be zero? I know that often in the literature it is understood that 0 and 1 are "coprime", so if we consider for example a $(0,1)$-cabling of the unknot with $d=2$, we get the two-components unlink (which is split). Also, it seems to me that these "extreme" cases are basically the only way to obtain a split solvable link. I would like to know if these cases are allowed, and so if solvable links can be split. I wasn't able to find many references about solvable links; hence, I appreciate also general references about solvable links.