Given a knot, we ask how its Khovanov and Khovanov-Rozansky homologies changeunder the operation of introducing twists in a pair of strands. We obtain longexact sequences in homology and further algebraic structure which is then usedto derive topological and computational results. Two of our applicationsinclude giving a new way to generate arbitrary numbers of knots with isomorphichomologies and finding an infinite number of mutant knot pairs with isomorphicreduced homologies.
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