Quantum (sl_n, \land V_n) link invariant and matrix factorizations

TL;DR

This paper generalizes Khovanov-Rozansky homology to define a new homology for quantum (sl_n,  V_n) link invariants, proving invariance for certain colored links and establishing new link invariants.

Contribution

It introduces a homology theory associated with quantum (sl_n,  V_n) invariants, extending previous categorifications to broader representations and colors.

Findings

Homology is a link invariant for [1,k]-colored diagrams.

Normalized Poincare polynomial is a link invariant for [i,j]-colored diagrams.

The work generalizes Khovanov-Rozansky homology to new quantum group representations.

Abstract

M. Khovanov and L. Rozansky gave a categorification of the HOMFLY-PT polynomial. This study is a generalization of the Khovanov-Rozansky homology. We define a homology associated to the quantum $(sl_n,\land V_n)$ link invariant, where $\land V_n$ is the set of the fundamental representations of the quantum group of $sl_n$. In the case of a [1,k]-colored link diagram, we prove that its homology is a link invariant. In the case of an [i,j]-colored link diagram, we define a normalized Poincare polynomial of its homology and prove the polynomial is a link invariant.