Virtual-knot problem 43 — In his wonderful paper , which firstly had the title “Knot Floer Homotopy”, Sarkar constructs a cell complex.

Definitions and mathematical statement

The source describes Sarkar's construction, initially titled "Knot Floer Homotopy," of a cell complex whose homology agrees with Floer homology, using the Manolescu--Ozsváth--Sarkar--Szabó--Thurston grid-diagram approach and shellability. It asks for an analogous construction for Khovanov homology, suggesting Bloom's construction as a more promising starting point than ordering the chains/states in the original Khovanov definition.

For a diagram $D$, write $\operatorname{Kh}^{i,j}(D;A)$ for the intended Khovanov groups, where $i$ is homological and $j$ quantum grading and $A$ is the chosen coefficient ring. A precise homological realization might seek combinatorially defined based cell or simplicial complexes $X_j(D)$, with stated shifts $s_j$, satisfying \[\widetilde H_{i+s_j}(X_j(D);A)\cong\operatorname{Kh}^{i,j}(D;A),\] or a specified equivalent realization appropriate to the chosen grading convention. The shifts/reduced conventions are necessary formulation choices rather than a claim that ordinary unshifted homology realizes negative chain degrees. If a cohomological or stable-spectrum realization is intended instead, its relationship to the source's homology/cell-complex wording must be explained rather than silently substituted.

The construction should specify the combinatorial cells or simplices and attachment/incidence data, identify the resulting chain theory with the chosen Khovanov theory, and state the resulting diagram-invariance property at the homology and, if claimed, homotopy/stable levels. The excerpt gives no actual ordering, poset, shelling or attachment rule and does not assert that every chain complex automatically has such a realization. The precise Bloom construction, Khovanov version and scope of diagrams are absent. Although the record belongs to a virtual-knot survey and the research summary mentions all virtuals, the quoted request itself does not explicitly restrict to or quantify over all virtual links; that extension requires a specified virtual Khovanov convention.

Short English statement

Construct a combinatorial cell or simplicial realization of Khovanov homology analogous to Sarkar's Floer construction, potentially using Bloom's model. The grading, homology-versus-cohomology and classical-versus-virtual scope must be fixed.

Sources

• Local UnsolvedMath registry, permanent ID $10600043$, exact source code catalogue entry.
• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory, revised survey, problem-list item 43: https://arxiv.org/abs/1409.2823. Registry access 2026-07-29; source-tex extraction. Embedded citations for Sarkar and Bloom are elided. "Knot Floer Homotopy" is reported as an initial title, not a verified final bibliographic title.
• The source names the Manolescu--Ozsváth--Sarkar--Szabó--Thurston grid approach. The research summary attributes substantial analogous progress to Lipshitz--Sarkar and mentions thickened-surface extensions, without precise citations.