Kirby Problem 1.34

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. Triply graded HOMFLYPT homology refers to the Khovanov--Rozansky braid-defined theory whose graded Euler characteristic is the HOMFLYPT polynomial. Its unreduced, middle, reduced and totally reduced variants differ by basepoint reduction; totally reduced means reduction at a basepoint on every component. Functoriality means assigning maps to oriented link cobordisms, invariant under the appropriate equivalences and compatible with identity cobordisms and composition; for reduced variants the cobordisms carry paths joining the basepoints. Cautis' theories here mean the braid-defined HOMFLYPT categorifications colored by arbitrary partitions and the finite-dimensional $\mathfrak{sl}(N)$ categorifications colored by symmetric powers, identified in the original remarks.

Statement recorded in the supplied source.

(a) Construct a description of Khovanov--Rozansky triply graded HOMFLYPT homology from arbitrary planar link diagrams, invariant under all Reidemeister moves, and determine whether some flavor admits functorial link-cobordism maps. (b) Obtain the same planar-diagram description and determine the same functoriality question for the link homologies of Cautis specified above. [S1][S2]

Short English statement

Replace the braid-only definitions of the specified HOMFLYPT and Cautis theories by arbitrary planar diagrams and determine which variants extend to cobordism functors.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2693 (KP-1.34). 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 1.34 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.