Concordance invariants from $U(1) \times U(1)$-equivariant Khovanov homology

TL;DR

This paper introduces new concordance invariants derived from a specialized equivariant Khovanov homology, expanding the algebraic framework and analyzing its properties under mirroring.

Contribution

It develops $U(1) imes U(1)$-equivariant Khovanov homology and extracts novel concordance invariants using algebraic filtrations, also extending the reduced theory and examining mirroring effects.

Findings

Two families of concordance invariants are constructed.

The behavior of the reduced theory under mirroring is analyzed.

The algebraic filtrations lead to new insights into knot concordance.

Abstract

We study Khovanov homology over the Frobenius algebra $\mathbb{F}[U,V,X]/((X-U)(X-V))$, or $U(1) \times U(1)$-equivariant Khovanov homology, and extract two families of concordance invariants using the algebraic $U$-power and $V$-power filtrations on the chain complex. We also further develop the reduced version of the theory and study its behavior under mirroring.