Colored sl(N) homology and SU(N) representations

Abstract

We provide the first complete computations of colored $\mathrm{sl}\left(N\right)$ homology for a nontrivial knot. In doing so, we show that the colored $\mathrm{sl}\left(N\right)$ homology of the trefoil labeled by an exterior power of the defining representation is isomorphic to the cohomology of a closed manifold naturally associated to the trefoil. This manifold is the set of homomorphisms from the fundamental group of the complement of the trefoil to $\mathrm{SU}\left(N\right)$ that send meridians to a particular conjugacy class depending on the label. We also provide complete computations and analogous isomorphisms for the first nontrivial link, the Hopf link.