I wonder if there is an example of a knot $K$ in the 3-sphere which can be realized as cables of two distinct (up to isotopy) knots $K\_1 \neq K\_2$.


It is known that if a knot $K$ is the $(p,q)$-cable of another knot $K'$, then there is a unique annulus in its exterior $X\_K$ with slope $pq$ on $\partial X\_K$. This is the only restriction on possible cabling configurations of $K$ that I am aware of. Can we somehow argue that $K'$ is uniquely determined by $K$?



Can a knot be cables of two different knots?
Yes, $K'$ is uniquely determined by $K$ and so, no, a knot cannot be a cable in two different ways. This follows from the [Gordon-Luecke theorem](https://en.wikipedia.org/wiki/Gordon%E2%80%93Luecke_theorem) and from a result of [Feustel and Whitten](https://cms.math.ca/openaccess/cjm/v30/cjm1978v30.1284-1295.pdf). See Lemma 5 in this paper of [Malyutin](https://arxiv.org/pdf/1612.03368.pdf).


