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.


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