Kirby Problem 1.30

Definitions and mathematical statement

Let $K\subset S^3$ be a smooth knot. Use unreduced integral Khovanov homology $\mathrm{Kh}^{i,j}(K;\mathbb Z)$, a bigraded finitely generated abelian group. It is obtained from the cube of the two smoothings at each crossing of a diagram: a resolution with $r$ circles receives the tensor power $A^{\otimes r}$, where $A=\mathbb Z[X]/(X^2)$. Edges of the cube use multiplication for merging circles and comultiplication $\Delta(1)=1\otimes X+X\otimes1$, $\Delta(X)=X\otimes X$ for splitting them. The usual signs and grading shifts give the link-invariant homology.

The conjecture states that, whenever $K$ is not isotopic to the unknot, there exist gradings $i,j$ and a nonzero element $x$ with \[ x\in\mathrm{Kh}^{i,j}(K;\mathbb Z),\qquad 2x=0. \] Thus the integral group must contain an element of order exactly two. It is essential to retain integer coefficients: working over a field alone loses the torsion being asked for. The assertion applies to all nontrivial knots, without an alternating, positive, or crossing-number restriction.

Short English statement

Does every nontrivial knot have an element of order two in its integral Khovanov homology?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2689 (KP-1.30), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 1.30. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.