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A ** two-component link** in $\mathbb{RP}^3$ is any embedding $S^1\uplus S^1\to \mathbb{RP}^3$. Two such links are

**if there exists a homotopy between the maps such that the images of the circles remain disjoint at each intermediate stage. (Note that each circle may pass through**

*homotopic**itself*during the homotopy.)

What is the set of homotopy classes of such links?

If either of the circles is nontrivial in $\pi_1(\mathbb{RP}^3)$, then it seems to me that the possibilities are completely determined by the linking numbers of the various components of the preimages in $S^3$. However, if both of the circles are nullhomotopic then the situation in the cover is much more complicated.