Quantum topological Hochschild homology and annular Khovanov spectra

TL;DR

This paper introduces a new stable homotopy refinement of quantum annular Khovanov homology, connecting it with quantum topological Hochschild homology and unifying previous constructions in the field.

Contribution

It constructs a novel stable homotopy refinement of quantum annular Khovanov homology and demonstrates its equivalence with quantum topological Hochschild homology of spectral tangle bimodules.

Findings

The new refinement agrees with qTHH of spectral Chen-Khovanov tangle bimodules.

It recovers earlier constructions of Krushkal and collaborators.

Establishes a link between quantum Hochschild homology and stable homotopy theory.

Abstract

Topological Hochschild homology is a topological analogue of classical Hochschild homology of algebras and bimodules. Beliakova, Putyra, and Wehrli introduced quantum Hochschild homology (qHH) and used it to define a quantization of annular Khovanov homology as qHH of the tangle bimodules of Chen-Khovanov and Stroppel. After introducing quantum topological Hochschild homology (qTHH), we construct a new stable homotopy refinement of quantum annular Khovanov homology and show that it agrees with qTHH of the spectral Chen-Khovanov tangle bimodules of Lawson, Lipshitz, and Sarkar. We also show that this new stable homotopy refinement recovers the construction introduced in earlier work of Krushkal together with the first and third authors.