Khovanov-type homologies of null homologous links in $\mathbb{RP}^3$

TL;DR

This paper introduces a new family of Khovanov-type homologies for null homologous links in real projective 3-space, unifying categorifications of the Kauffman bracket and linking to Heegaard Floer homology.

Contribution

It defines a generalized homology theory depending on an input parameter, recovering known invariants and establishing a spectral sequence to Heegaard Floer homology.

Findings

Recovers Asaeda-Przytycki-Sikora's categorification of the Kauffman bracket.

Constructs a spectral sequence converging to Heegaard Floer homology.

Provides a unified framework for link invariants in $\,\mathbb{RP}^3$.

Abstract

Let L be a null homologous link in $\mathbb{RP}^3$. We define Khovanov-type homologies of L which depend on an extra input $\alpha = (V_0,V_1,f,g)$ consisting of two graded vectors spaces and two maps between them. With some specific choice of $\alpha = \alpha_{APS}$, we recover the categorification of the Kauffman bracket due to Asaeda-Przytycki-Sikora. With another choice of $\alpha = \alpha_{HF}$, we construct a spectral sequence from our theory converging to the Heegaard Floer homology of the even branched double cover of $\mathbb{RP}^3$.