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Khovanov Homology for Alternating Tangles

We describe a “concentration on the diagonal” condition on the Khovanov complex of tangles, show that this condition is satisfied by the Khovanov complex of the single crossing tangles (!) and ("), and prove that it is preserved by alternating planar algebra compositions. Hence, this condition is satisfied by the Khovanov complex of all alternating tangles. Finally, in the case of 0-tangles, meaning links, our condition is equivalent to a well known result [Lee1] which states that the Khovanov homology of a non-split alternating link is supported in two diagonals. Thus our condition is a generalization of Lee’s Theorem to the case of tangles