Nonstabilized Weakly Reducible Heegaard Splittings

Definitions and mathematical statement

A Heegaard splitting of a compact orientable three-manifold is $M=C_1\cup_\Sigma C_2$, where $C_1,C_2$ are compression bodies with common positive boundary $\Sigma$. It is weakly reducible if there are compressing disks $D_i\subset C_i$ whose boundary curves on $\Sigma$ are disjoint. It is stabilized if there are compressing disks on opposite sides whose boundary curves meet transversely in exactly one point. Boundary stabilization is the usual amalgamation with a standard splitting in a collar of a boundary component; it is a distinct operation from ordinary stabilization.

The source asks two questions.

1. For integers $m$ with sufficiently large $|m|$, let $K_m=T(7,17,10m-4)$ be the twisted torus knot in the report and let $M_m$ be its complement, interpreted as the compact knot exterior when discussing Heegaard splittings. The source specifies a unique genus-two Heegaard splitting of $M_m$ and the weakly reducible splitting $\Sigma_m^{\partial}$ obtained from it by boundary stabilization. Is $\Sigma_m^{\partial}$ stabilized? The intended large-parameter assertion asks about all integers beyond an unspecified absolute-value threshold; no value of that threshold or assertion for small $m$ is supplied. The twisted-torus notation and the precise boundary-stabilization construction must be recovered from the report.
2. More generally, does there exist a three-manifold with exactly one boundary component admitting a weakly reducible, nonstabilized Heegaard splitting whose weak reduction is nonessential?

A weak reduction compresses $\Sigma$ along a disjoint opposite-side disk pair and then interprets the resulting surface according to the relevant conventions. Essential surfaces are ordinarily incompressible and not boundary parallel, with sphere and trivial-component exclusions specified separately. The extraction does not state whether "nonessential" means the entire resulting surface has no essential component, that some component is nonessential, or another convention. Nor does it specify whether one weak reduction or every weak reduction is meant. The second question is therefore retained with this explicit ambiguity; an existential choice of disks is not silently substituted for a universal one.

The uniqueness and weak reducibility in the first question are supplied features of the proposed examples, not independently proved claims in this draft. No assumption of irreducibility or incompressible boundary is added to the general existence question without source confirmation.

Short English statement

The first question asks whether boundary-stabilized genus-two splittings of the named twisted-torus-knot exteriors are ordinarily stabilized for large $|m|$. The second asks for a one-boundary-component example that is weakly reducible and nonstabilized but has a nonessential weak reduction, whose precise meaning needs clarification.

Sources

• Local UnsolvedMath registry, permanent ID 30000336, source code catalogue entry, "Nonstabilized Weakly Reducible Heegaard Splittings." Proposed year 2005; proposer unspecified.
• Low-Dimensional Manifolds (2006), Oberwolfach Reports, pp. 2519--2570, DOI 10.4171/owr/2005/45; https://doi.org/10.4171/owr/2005/45. The displayed report year 2006 differs from the DOI and proposed year 2005.
• Statement status corrected_verified; the supplied note says editorial and LaTeX cleanup preserved meaning and added no source-dependent content.
• Literature triage checked 2026-08-21: open. It says the passage explicitly leaves existence of the one-boundary-component example open. It gives no separate current assessment of each knot-family splitting. Created 2026-08-25; updated 2026-09-16. Time-sensitive status and source claims require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.