Kirby Problem 3.66

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.

Put $R=\mathbb Z[A,A^{-1}]$ and $R_{\mathbb Q}=\mathbb Q[A,A^{-1}]$. For a closed oriented $3$-manifold $Y$, the Kauffman bracket skein module $S(Y)$ is the free $R$-module on isotopy classes of unoriented framed links, including the empty link, modulo the local bracket relations \[ [L_\times]=A[L_0]+A^{-1}[L_\infty],\qquad [L\sqcup O]=-(A^2+A^{-2})[L]. \] The crossing and its two smoothings are the usual ordered Kauffman bracket local pictures in a ball, and $O$ is a disjoint zero-framed trivial circle. Let $S_{\mathbb Q}(Y)=S(Y)\otimes_R R_{\mathbb Q}$. The source calls this module tame if it is a direct sum of cyclic $R_{\mathbb Q}$-modules and there exists an odd integer $N\ge1$ for which it has no submodule isomorphic to $R_{\mathbb Q}/(\Phi_{2N}(A))$, where $\Phi_{2N}$ is the cyclotomic polynomial. Let $\operatorname{HP}^*_{\#}(Y)$ be Abouzaid--Manolescu's framed sheaf-theoretic $\operatorname{SL}_2(\mathbb C)$ Floer cohomology: it is the hypercohomology of the perverse sheaf of vanishing cycles for the framed complex-Lagrangian intersection associated to a Heegaard splitting, on $\operatorname{Hom}(\pi_1(Y),\operatorname{SL}_2(\mathbb C))$. Its degree-zero group is $\operatorname{HP}^0_{\#}(Y)$; take its dimension after tensoring its integral coefficients with $\mathbb C$. [S3]

Statement recorded in the supplied source.

For every closed oriented $3$-manifold $Y$:

1. Is \[ \dim_{\mathbb Q(A)}\left(S(Y)\otimes_R\mathbb Q(A)\right) =\dim_{\mathbb C}\left(\operatorname{HP}^0_{\#}(Y)\otimes_{\mathbb Z}\mathbb C\right)? \]
2. Characterize exactly when $S_{\mathbb Q}(Y)$ is tame in the sense above.

[S2]

Short English statement

Does generic Kauffman bracket skein rank equal degree-zero framed complex Floer dimension for every closed oriented three-manifold, and which Laurent-polynomial skein modules are tame?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2864 (KP-3.66). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (editors), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), Problem 3.66, p. 179 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 Mohammed Abouzaid and Ciprian Manolescu, A sheaf-theoretic model for SL(2,C) Floer homology, Journal of the European Mathematical Society 22 (2020), 3641--3695; arXiv:1708.00289v2, introduction, Theorem 1.3 and the framed hypercohomology definition. https://arxiv.org/abs/1708.00289v2.