A colored $\mathfrak{sl}(N)$ homology for links in $S^{3}$
Volume 499 / 2014
Dissertationes Mathematicae 499 (2014), 1-217
MSC: Primary 57M27.
DOI: 10.4064/dm499-0-1
Abstract
Fix an integer $N\geq 2$. To each diagram of a link colored by $1,\dots,N$ 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. The homology of this chain complex decategorifies to the Reshetikhin–Turaev $\mathfrak{sl}(N)$ polynomial of links colored by exterior powers of the defining representation.
