Annular Link Homology Theories and their Homotopical Refinements

Homology theories of links which categorify quantum link invariants have been developed over the past twenty years, starting with Khovanov's seminal categorification of the Jones polynomial. This thesis focuses on links in the thickened annulus and develops annular link homology in two main directions. First, we present joint work with Krushkal and Willis in which a stable homotopy refinement of Beliakova-Putyra-Wehrli's quantum annular homology is constructed. Second, we introduce equivariant annular link homology. This is comprised of sl(2) annular homology via filtrations, joint work with Khovanov in the sl(2) and sl(3) setting via foam evaluation and universal construction, and a treatment of the general gl(N) setting via foams.