2-cabling and tangle operators in Khovanov theory

TL;DR

This paper introduces an operator on 4-ended tangles derived from 2-cabling of strongly invertible knots, connecting Khovanov theory, Fukaya categories, and concordance invariants.

Contribution

It defines a new operator in tangle Khovanov theory induced by 2-cabling, linking knot invariants with categorical and geometric structures.

Findings

Describes the operator on 4-ended tangles and its restriction to cap-trivial tangles.

Establishes connections between tangle operators and Fukaya categories.

Provides geography results related to a recent concordance invariant.

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 $\mathcal{B}$, 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.