Khovanov homology in characteristic two and involutive monopole Floer homology

TL;DR

This paper constructs a spectral sequence linking characteristic two Khovanov homology of a link's mirror to an involutive monopole Floer homology of its branched double cover, revealing new connections in low-dimensional topology.

Contribution

It establishes a spectral sequence from Bar Natan's characteristic two Khovanov homology to an involutive monopole Floer homology, extending the understanding of link invariants in Floer theory.

Findings

Existence of a spectral sequence converging to involutive monopole Floer homology.

The E^2-page is related to characteristic two Khovanov homology of the mirror link.

Conjecture of a similar spectral sequence in the Pin(2)-monopole Floer homology setting.

Abstract

We study the conjugation involution in Seiberg-Witten theory in the context of the Ozsv\'ath-Szab\'o and Bloom's spectral sequence for the branched double cover of a link $L$ in $S^3$. We prove that there exists a spectral sequence of $\mathbb{F}[Q]/Q^2$-modules (where $Q$ has degree $-1$) which converges to $\widetilde{\mathit{HMI}}_*(\Sigma(L))$, an involutive version of the monopole Floer homology of the branched double cover, and whose $E^2$-page is a version of Bar Natan's characteristic two Khovanov homology of the mirror of $L$. We conjecture that an analogous result holds in the setting of $\mathrm{Pin}(2)$-monopole Floer homology.