Kirby Problem 1.37

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. Fix a complex simple Lie algebra $\mathfrak g$ and color each oriented link component by a finite-dimensional irreducible representation. Let $\mathrm{RT}_{\mathfrak g}(L;q)$ be the corresponding normalized Reshetikhin--Turaev quantum link invariant. A bigraded categorification has groups $H^{i,j}$ whose graded Euler characteristic is $\sum_{i,j}(-1)^i q^j\operatorname{rank}H^{i,j}$. Functoriality requires compatible maps for colored oriented cobordisms and compatibility with identity and composition. The requested odd version is an integral variant whose reduction modulo $2$ agrees with that of the original theory, analogous to odd Khovanov homology.

Statement recorded in the supplied source.

(a) For every link, every simple $\mathfrak g$ and every irreducible coloring, construct a bigraded link-homology theory functorial under link cobordisms with Euler characteristic $\mathrm{RT}_{\mathfrak g}(L;q)$. (b) For each theory construct an odd version with the same mod-$2$ reduction, or identify an obstruction to such an odd construction. [S1][S2]

Short English statement

Construct functorial colored categorifications for every simple Lie algebra, then construct odd variants or determine what prevents them.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2696 (KP-1.37). 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 1.37 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.