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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 $K$?

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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 and from a result of Feustel and Whitten. See Lemma 5 in this paper of Malyutin.

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