Kirby Problem 3.65

Definitions and mathematical statement

Let $Y$ be a closed connected oriented three-manifold, and put $R=\mathbb Z[A,A^{-1}]$. Its Kauffman bracket skein module $S_{2,\infty}(Y)$ is the free $R$-module on isotopy classes of unoriented framed links in $Y$, including the empty link, modulo two local relations: \[ [L_\times]=A[L_0]+A^{-1}[L_\infty],\qquad [L\sqcup U]=-(A^2+A^{-2})[L]. \] In the first relation the three links agree outside a ball, and inside it a framed crossing is replaced by its two planar smoothings in the usual Kauffman bracket convention. In the second, $U$ is an isolated zero-framed unknot bounding a disk disjoint from $L$.

Extend coefficients to the rational-function field $\mathbb Q(A)$. The generic rank is the dimension of \[ S_{2,\infty}(Y)\otimes_R\mathbb Q(A). \] Suppose $Y$ is prime, meaning a connected-sum decomposition of $Y$ has a summand diffeomorphic to $S^3$. If this dimension is one, must $Y$ be diffeomorphic to $S^3$ or $S^1\times S^2$?

The rank condition is over the field of rational functions in an indeterminate. It is not a specialization at one complex value of $A$, and it does not impose finite generation over $R$. In particular, torsion submodules may disappear upon extending coefficients and are not excluded by the hypothesis. The conclusion concerns the underlying manifold without any link chosen in advance.

Short English statement

Does generic skein rank one recognize the sphere and the sphere-times-circle among prime three-manifolds?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2863 (KP-3.65), 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 3.65. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.