A colored sl(N)-homology for links in S^3

TL;DR

This paper constructs a new colored sl(N)-homology for links in S^3, extending Khovanov-Rozansky homology, and proves its invariance under Reidemeister moves, connecting it to the Reshetikhin-Turaev polynomial.

Contribution

It introduces a novel chain complex of graded matrix factorizations for colored links, generalizing existing homology theories and establishing invariance under Reidemeister moves.

Findings

Homotopy invariance under Reidemeister moves

Decategorification yields the Reshetikhin-Turaev sl(N) polynomial

Special case recovers Khovanov-Rozansky homology

Abstract

Fix an integer N>1. To each diagram of a link colored by 1,...,N, we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain complex defined by Khovanov and Rozansky in arXiv:math/0401268. The homology of this chain complex decategorifies to the Reshetikhin-Turaev sl(N) polynomial of links colored by exterior powers of the defining representation.