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 $\alpha ={\alpha }_{APS}$, 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 ${\mathbb{RP}}^3$.

32 pages, 14 figures