Let $$L = K_1 \cup K_2 \subset S^3$$ be a two component framed link with $$lk(K_1,K_2) = 1$$. Let $$\hat{L}$$ denote the set of all links obtained by handlesliding $$L$$ around an an arbitrary number of times. Is there a knot contained in a link in $$\tilde{L}$$ with Arf invariant 1?
For example, if $$L$$ is a 0-framed Hopf link, we can change crossings via handleslides, and thus every knot is contained in a link in $$\hat{L}$$.
Note that without the linking number condition this is not true: for example, if $$L$$ is a two component unlink with $$0$$-framing, then every knot contained in a link in $$\hat{L}$$ is ribbon, and hence has vanishing Arf invariant.