Kirby Problem 1.36

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.

For an oriented surface $\Sigma$, the relevant skein algebra is the module of framed links in $\Sigma\times[0,1]$ modulo the local $\mathfrak{sl}(2)$, $\mathfrak{sl}(N)$ or HOMFLYPT skein relations, with multiplication by stacking in the interval direction. In the $\mathfrak{sl}(2)$ case the Kauffman bracket relations resolve a crossing as $A$ times one smoothing plus $A^{-1}$ times the other, and remove a trivial circle with factor $-A^2-A^{-2}$. The $\mathfrak{sl}(N)$ and HOMFLYPT variants use their corresponding oriented quantum skein relations. A categorification should assign graded link objects whose Grothendieck classes give the skein algebra, together with a categorical multiplication inducing the stacking product; merely assigning link homologies without a compatible product does not fulfill the target.

Statement recorded in the supplied source.

Construct such categorifications of the $\mathfrak{sl}(2)$, $\mathfrak{sl}(N)$ and HOMFLYPT skein algebras for surfaces, including a categorical analogue of the skein multiplication. [S1][S2]

Short English statement

Categorify all three specified surface skein algebras, with the stacking product represented categorically.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2695 (KP-1.36). 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 1.36 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.