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Khovanov-type homologies of null homologous links in $\mathbb{RP}^3$
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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 $\mathbb{RP}^3$.
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Link in $\mathbb{R}\mathbb{P}^3$ and the Topological Vertex
First topological-vertex computations of colored unknot and Hopf link invariants in RP^3 yield power series with positive integer q-expansions, conjecturally the Poincare series of a new link homology.
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