Khovanov homology for alternating tangles

TL;DR

This paper introduces a diagonal concentration condition for Khovanov complexes of tangles, proves its preservation under alternating planar algebra compositions, and generalizes Lee's Theorem to tangles.

Contribution

It defines a new diagonal concentration condition, shows it holds for all alternating tangles, and extends Lee's Theorem from links to tangles.

Findings

The condition is satisfied by the Khovanov complex of single crossing tangles.

The condition is preserved under alternating planar algebra compositions.

For links, the condition aligns with known support on two diagonals in Khovanov homology.

Abstract

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 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 links, our condition is equivalent to a well known result which states that the Khovanov homology of a non-split alternating link is supported on two diagonals. Thus our condition is a generalization of Lee's Theorem to the case of tangles