Kirby Problem 1.28

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.

Let $I=[0,1]$ and $L\subset S^1\times I\times I$ be an annular link. Its wrapping number $w(L)$ is the minimum transverse geometric intersection number with a meridional disk, over isotopies. The Kauffman bracket skein module of the solid torus has basis $1,z,z^2,\ldots$, where $z$ is an essential core circle. Write $\langle L\rangle=\sum_k f_k(A)z^k$ and define its annular degree as $\max\{k:f_k\ne0\}$. Annular Khovanov homology $\mathrm{AKh}^{i,j,k}(L)$ is the associated-graded homology for the annular filtration: in each resolution an essential circle labeled $v_+$ or $v_-$ contributes $+1$ or $-1$ to $k$, while contractible circles contribute zero. Use the conventional integral theory.

Statement recorded in the supplied source.

(a) Is $w(L)=\max\{k:f_k(A)\ne0\}$? (b) Is $w(L)=\max\{k:\mathrm{AKh}^{i,j,k}(L)\ne0\text{ for some }i,j\}$? [S1][S2]

Short English statement

Do the annular Kauffman bracket and annular Khovanov grading each detect the minimum geometric winding of a link?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2687 (KP-1.28). 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 1.28 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.