  Given a band sum of a split two-component link along a nontrivial band, we
obtain a family of knots indexed by the integers by adding any number of full
twists to the band. We show that the knots in this family have the same
Heegaard knot Floer homology and the same instanton knot Floer homology. In
contrast, a generalization of the cosmetic crossing conjecture predicts that
the knots in this family are all distinct. We verify this prediction by showing
that any two knots in this family have distinct Khovanov homology. Along the
way, we prove that each of the three knot homologies detects the trivial band.
