Khovanov-type homologies of null homologous links 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 $\alpha ={\alpha }_{\mathrm{APS}}$, we recover the categorification of the Kauffman bracket due to Asaeda–Przytycki–Sikora. With another choice of $\alpha ={\alpha }_{\mathrm{HF}}$, we construct a spectral sequence from our theory converging to the Heegaard–Floer homology of the even branched double cover of ${\mathbb{RP}}^3$.