Quantum topological Hochschild homology and annular Khovanov spectra

TL;DR

This work defines quantum topological Hochschild homology (qTHH) for graded $ extbf{S}[G]$-algebras and uses it to lift quantum annular Khovanov invariants to stable homotopy types. The authors construct the quantum annular spectrum $ ext{X}_{ ext{A}_q}(L)$ with a $G$-action and prove a central equivalence: qTHH of spectral Chen–Khovanov tangle bimodules recovers $ ext{X}_{ ext{A}_q}(L)$, generalizing prior algebraic lifts. They show that for finite cyclic $G$, $ ext{X}_{ ext{A}_q}(L)$ agrees with the AKW quantum annular spectrum, tying new spectral constructions to established invariants. The paper develops a comprehensive categorical framework—shape multicategories, divided cobordisms, Burnside 2-categories, and rectification—to lift Khovanov-type data from chain complexes to spectra, culminating in a robust equivalence between spectral and algebraic quantum annular invariants with potential $ ext{U}_q(rak{sl}_2)$-actions and rich representation-theoretic structure. The results provide a concrete pathway from tangle bimodules to equivariant spectra, enabling computation of qTHH via BPW’s and AKW’s frameworks and offering a new toolkit for spectral link invariants in the annular setting.

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.

Figures (11)

• Figure 1: A schematic of various Khovanov invariants. The first column consists of scalar extended bimodule invariants associated to tangles. The second column consists of quantum Hochschild constructions. The third column consists of annular link invariants. We use $\longmapsto$ to denote an assignment of one object to another (such as taking homology of a chain complex), $\longrightarrow$ to denote an actual morphism between objects, and $\overset{\simeq}{\longleftrightarrow}$ to denote a zig-zag of weak equivalences. The portions in red indicate new constructions in this paper.
• Figure 2: The local Bar-Natan relations.
• Figure 3: The two resolutions of a crossing.
• Figure 4: A depiction of Definition \ref{['def:matchings and platforms']}
• Figure 5: A planar $(2,6)$-tangle $T$, elements $a \in B^{2,0}$ and $b\in B^{4,2}$, and the closure $a T \widehat{b}$. The dashed arcs in $a T \widehat{b}$ are added to close the picture to a union of circles in the plane. Here $k=0$ and $h(k)=2$ in the notations of Equation \ref{['eq:FCKBar(T,k)']}. The type of each circle is indicated.

Theorems & Definitions (181)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Definition 2.7
• Definition 2.8