Colored sl(N) homology and SU(N) representations of the trefoil

Joshua Wang

Published 2022-11-15Version 1

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 prove the analogous isomorphisms for the Hopf link. These isomorphisms suggest a close relationship between colored sl(N) homology and instanton Floer homology with gauge group SU(N).