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

TL;DR

This paper computes the colored sl(N) homology for the trefoil knot, revealing an isomorphism with the cohomology of a manifold related to SU(N) representations, advancing understanding of knot invariants and their geometric counterparts.

Contribution

It provides the first complete calculations of colored sl(N) homology for a nontrivial knot and establishes a novel link to SU(N) representation varieties.

Findings

Colored sl(N) homology of the trefoil matches the cohomology of a specific SU(N) representation manifold.

Complete computations for the trefoil and Hopf link are provided.

Isomorphisms between homology and representation manifold cohomology are established.

Abstract

We provide the first complete computations of colored sl(N) homology for a nontrivial knot. In doing so, we show that the colored sl(N) 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 SU(N) 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.