Kirby Problem 3.61

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.

A balanced sutured manifold $(M,\gamma)$ is a compact oriented $3$-manifold without closed components, with oriented sutures in annular neighborhoods $\gamma\subset\partial M$, whose complement is $R_+(\gamma)\sqcup R_-(\gamma)$, with compatible boundary orientations, $\chi(R_+)=\chi(R_-)$, and every boundary component meeting $\gamma$. The group $\operatorname{SFH}(M,\gamma;k)$ is sutured Heegaard Floer homology, the homology of its admissible sutured Heegaard chain complex. The group $\operatorname{SHI}(M,\gamma;\mathbb C)$ is sutured instanton homology: the distinguished surface-eigenspace of instanton Floer homology of an admissible closure of the sutured manifold. Their isomorphism classes are independent of diagram and closure choices. For a closed genus-two surface $\Sigma\subset\operatorname{int}M$, let $\iota$ be its hyperelliptic involution, the involution with quotient $S^2$ and six branch points. The genus-two mutant $M_\Sigma$ is obtained by cutting along $\Sigma$ and regluing by $\iota$, leaving boundary sutures unchanged. For a knot disjoint from $\Sigma$, $K_\Sigma$ denotes its image in the mutant. All dimensions below are total, ungraded dimensions over $\mathbb F_2=\mathbb Z/2\mathbb Z$.

Statement recorded in the supplied source.

Is genus-two mutation dimension preserving for Heegaard Floer homology? In the broad sutured formulation specified by the source, is \[ \dim_{\mathbb F_2}\operatorname{SFH}(M,\gamma;\mathbb F_2) =\dim_{\mathbb F_2}\operatorname{SFH}(M_\Sigma,\gamma;\mathbb F_2) \] for every balanced $(M,\gamma)$ and every such embedded $\Sigma$? In particular, ask the same equality for total $\widehat{\operatorname{HF}}$ of closed $3$-manifolds and total $\widehat{\operatorname{HFK}}$ of knots disjoint from the mutation surface. Equality in each spin$^c$, Alexander or other grading is not required. [S2]

Short English statement

Does genus-two mutation preserve the total dimension of sutured Heegaard Floer homology, and hence the total hat Floer dimensions of closed three-manifolds and knots?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2859 (KP-3.61). 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 3.61, p. 174 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.