Can we split a splittable link by applying Reidemeister moves to non-self crossings only

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Suppose $L=K_1 \cup K_2$ is a link consisting of two components $K _1$ and $K_2$ With some non-self crossings between $K _1$ and $K_2$ . Suppose $L$ is isotopic to the unlink, then can we split the two components by just applying Reidemeister moves to the crossings between $K _1$ and $K_2$ and keep the self crossings unchanged ?