  Given a knot, we ask how its Khovanov and Khovanov-Rozansky homologies change
under the operation of introducing twists in a pair of strands. We obtain long
exact sequences in homology and further algebraic structure which is then used
to derive topological and computational results. Two of our applications
include giving a new way to generate arbitrary numbers of knots with isomorphic
homologies and finding an infinite number of mutant knot pairs with isomorphic
reduced homologies.
