Étale Homotopy Realization of Khovanov Homology

Definitions and mathematical statement

Write $\operatorname{Kh}(K)$ for the Khovanov homology of a link $K$. The supplied record does not specify the ambient space and orientation conventions for links, the coefficient ring, the reduced or unreduced version of Khovanov homology, or the grading conventions. These choices must be fixed before a precise realization statement can be formulated.

The source asks whether, for every link $K$ in the intended class, one can associate an appropriate étale homotopy type $E(K)$ that represents $\operatorname{Kh}(K)$ and is itself a link invariant. Schematically, the requirements are \[ K\sim K'\ \Longrightarrow\ E(K)\simeq E(K'), \qquad \operatorname{Rec}(E(K))\cong\operatorname{Kh}(K). \] Here $\sim$ denotes the intended equivalence of links, $\simeq$ denotes the intended equivalence of étale homotopy types, and $\operatorname{Rec}$ is a placeholder for the required recovery operation. None of these choices is specified sufficiently in the supplied record. In particular, the record does not identify the site, topos, or other object whose étale homotopy type is intended, nor whether representation means recovery by homology, cohomology, or another construction, with which coefficients and gradings. The displayed conditions therefore express the source question schematically; they do not supply a fully determined realization conjecture.

Short English statement

The question asks whether every link admits an étale homotopy type that is invariant under the intended equivalence of links and recovers its Khovanov homology. The supplied record leaves the homotopy construction, recovery operation, and several conventions unspecified, so a precise conjecture requires further source clarification.

Sources

• Local UnsolvedMath registry, permanent ID $30002594$, source problem code catalogue entry, title: Étale Homotopy Realization of Khovanov Homology. This draft uses the supplied complete source record.
• Supplied report citation: Algebraic Structures in Low-Dimensional Topology (2015), Oberwolfach Reports, pp. 1403--1458, DOI 10.4171/owr/2014/26. https://doi.org/10.4171/owr/2014/26. The record gives proposed year 2014; the relationship between that year and the citation's 2015 date is unresolved here.
• Supplied literature-source attribution: Bryden--Kauffman, a topos-theoretic interpretation of Khovanov homology. https://doi.org/10.4171/owr/2014/26. The record supplies no individual proposer and no more specific contribution locator.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, describes the report as presenting the realization as a conjecture, and reports that no authoritative proof or disproof was located. This is supplied dated curation, not an independently verified literature assessment.
• The supplied registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report, with hypotheses and notation restored while preserving the open question. This is a supplied verification claim; the report and that claim have not been independently checked for this draft.