Kirby Problem 1.35

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

All knots and links below are smooth in the oriented $3$-sphere, up to ambient isotopy, unless another ambient manifold is specified. The integral symplectic Khovanov group $\mathrm{Kh}_{\mathrm{symp}}(K;\mathbb Z)$ is the Seidel--Smith Floer-theoretic invariant obtained from matching Lagrangians in transverse slices to nilpotent orbits. Write $\mathrm{Kh}(K;\mathbb Z)$ for ordinary integral Khovanov homology and $\mathrm{Kh}_{\mathrm{odd}}(K;\mathbb Z)$ for the odd integral theory of Ozsváth--Rasmussen--Szabó, whose circle-state algebra uses exterior rather than ordinary Frobenius multiplication. Grading comparisons use the conventional total grading in the symplectic construction.

Statement recorded in the supplied source.

(a) Is integral symplectic Khovanov homology isomorphic to integral ordinary Khovanov homology? (b) Construct an odd symplectic Khovanov theory, a Floer-theoretic counterpart of odd Khovanov homology, as requested in the original problem. [S1][S2]

Short English statement

Compare the symplectic and combinatorial theories over the integers and build a symplectic counterpart of odd Khovanov homology.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2694 (KP-1.35). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, author preliminary edition (2026), Problem 1.35 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.