Kirby Problem 1.31

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. For coprime nonzero integers $m,n$, $T(m,n)$ is the torus knot of those parameters; changing the sign of one parameter gives its mirror, and a parameter of absolute value one gives the unknot. Write $\mathrm{Kh}^{i,j}(K;\mathbb Z)$ for integral Khovanov homology, with homological and quantum gradings. Write $\mathrm{KhR}_{N}^{i,j}(K;\mathbb Z)$ for integral fundamental-representation $\mathfrak{gl}(N)$ Khovanov--Rozansky link homology, for $N\geq2$, using the integral link-homology constructions discussed in the source. A computation of these groups includes all graded free ranks and torsion summands, rather than only their Euler characteristics.

Statement recorded in the supplied source.

Determine (a) $\mathrm{Kh}^{i,j}(T(m,n);\mathbb Z)$ for every torus knot and all $i,j$, and (b) the corresponding $\mathfrak{gl}(N)$ Khovanov--Rozansky homologies for all torus knots and all $N\geq2$, including the torsion phenomena emphasized in the original remarks. [S1][S2]

Short English statement

Compute all graded homology groups, including integral torsion, for every torus knot in the Khovanov and general linear Khovanov--Rozansky theories.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2690 (KP-1.31). 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 1.31 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.