2-cabling and tangle operators in Khovanov theory
Abstract
We describe an operator on 4-ended tangles that is induced by 2-cabling of a strongly invertible knot. By passing to the 4-ended tangle Khovanov theory of Kotelskiy-Watson-Zibrowius, this induces an operator on the category of type D structures over the Bar-Natan algebra B, as well as on a Fukaya category of the 4-punctured 2-sphere. We provide a full description of this operator's restriction to cap-trivial tangles. Finally, we extract geography results that are inspired by a recent concordance invariant of Lewark-Zibrowius.
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