Stable homotopy refinement of quantum annular homology

TL;DR

This paper develops a stable homotopy refinement of quantum annular homology, providing a new spectrum-level perspective that enhances the understanding of link invariants in the annular setting.

Contribution

It introduces a stable homotopy refinement for quantum annular homology using an equivariant Burnside category approach, extending previous homology theories.

Findings

Constructs a $ extbf{Z}/r extbf{Z}$-equivariant spectrum for each annular link

Recovers the stable homotopy refinement of annular Khovanov homology via quotient

Studies spectrum-level properties of quantum annular homology

Abstract

We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq 2$ we associate to an annular link $L$ a naive $\mathbb{Z}/r\mathbb{Z}$-equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb{Z}[\mathbb{Z}/r\mathbb{Z}]$. The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.