This thesis formulates some new structures in the Khovanov theory of strongly invertible knots, via an associated tangle construction. Strongly invertible knots are those with a Z/2Z-symmetry given by an involution on S3 that fixes the knot set-wise but reverses a choice of orientation. Such a knot has a naturally associated tangle that serves as a key witness for the relationship between the Khovanov homology of a knot or link and the Heegaard Floer homology of its branched double cover. The structures in question are inspired by this relationship, and the work has connections to the smooth and smooth equivariant concordance groups. The first structure is a mapping cone formula that identifies a new tangle invariant N. This invariant is related to a concordance invariant recently defined by Lewark–Zibrowius. Our study leads to a proof of a conjecture that they made, via an analysis of bigraded Khovanov invariants of 2-component links. The second structure is 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, 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. This structure is a first step in a bimodule theory for Khovanov theory, similar to the theory of Lipshitz– Ozsv ́ath–Thurston for bordered Heegaard Floer homology.
Mihai Marian (Fri,) studied this question.