Quantum (sln,∧Vn) link invariant and matrix factorizations

Abstract

In this paper, we give a generalization of Khovanov-Rozansky homology. We define a homology associated to the quantum $({\mathfrak{sl}}_n,\wedge V_n)$ link invariant, where $\wedge V_n$ is the set of fundamental representations of $U_q\left({\mathfrak{sl}}_n\right)$. In the case of an oriented link diagram composed of $[k,1]$-crossings, we define a homology and prove that the homology is invariant under Reidemeister II and III moves. In the case of an oriented link diagram composed of general $[i,j]$-crossings, we define a normalized Poincaré polynomial of homology and prove that the normalized Poincaré polynomial is a link invariant.