Real monopoles and a spectral sequence from Khovanov homology

TL;DR

This paper introduces a new real monopole Floer homology for links, proves an exact triangle, relates it to a known invariant, and constructs a spectral sequence connecting it to Khovanov homology.

Contribution

It defines a novel real monopole Floer homology for links, establishes an exact triangle, and constructs a spectral sequence linking it to Khovanov homology.

Findings

Euler characteristic matches Miyazawa's invariant.

Spectral sequence abuts to the new homology with E2 page as reduced Khovanov homology.

Examples illustrate the properties of the new homology.

Abstract

Given a based link $(K,p)$, we define a "tilde"-version $\tilde{HMR}(K,p)$ of real monopole Floer homology and prove an unoriented skein exact triangle. We show the Euler characteristic of $\tilde{HMR}(K,p)$ is equal to Miyazawa's invariant $|deg(K)|$ arXiv:2312.02041 and examine some examples. Further, we construct a spectral sequence over $\mathbb{F}_2$ abutting to $\tilde{HMR}(K,p)$, whose $E_2$ page is the reduced Khovanov homology $Khr(\overline{K})$ of the mirror link $\overline{K}$.