The generalized cosmetic crossing conjecture for band surgeries

Let $L$ be a two-component link, let $b$ be a band, and let $K_b$ be the oriented knot obtained from $L$ by band surgery along $b$. For each $n\in\mathbf{Z}$, let $K_{b+n}$ be the oriented knot obtained from $K_b$ by adding $n$ full twists to $b$. If $L$ is split, call $b$ trivial when an embedded sphere splits $L$ and intersects $b$ in a single arc. Generalized cosmetic crossing conjecture. If distinct integers $n$ and $m$ satisfy that $K_{b+n}$ and $K_{b+m}$ are isotopic as oriented knots, then $L$ is split and $b$ is trivial. The conjecture predicts that, for a nontrivial band on a split link, the knots obtained by adding different numbers of full twists are pairwise non-isotopic; the paper verifies this prediction in the setting studied there using Khovanov homology.