Let M be a compact orientable simple 3-manifold, let F be a component of partial M with genus at least 2, and let alpha,beta be separating slopes on F. In the reducible–reducible case, if both Dehn fillings M(alpha) and M(beta) are reducible, then Delta(alpha,beta)=0; equivalently, alpha=beta. The full conjecture concerns the corresponding distance bounds for all separating slopes that yield degenerating Dehn fillings.

Qiu–Zhang conjecture on separating degenerating slopes

The conjecture concerns distance bounds for separating slopes that produce degenerating Dehn fillings. The full conjecture is not resolved.

Known results

Two separating reducing slopes satisfy $\Delta(\alpha,\beta)\leq 4$ on the relevant higher-genus boundary.
For torus boundary, Gordon and Luecke obtained $\Delta(\alpha,\beta)\leq 1$ for two reducing slopes.
In genus two, at most one separating slope produces a boundary-reducible handle addition.
Under additional boundary-compressibility hypotheses, the distance bound improves to $\Delta(\alpha,\beta)\leq 2$ .

October 2026 reducible–reducible case

Qilong Guo's preprint Reducible handle additions along separating slopes claims the previously conjectural reducible–reducible case without a boundary-genus restriction. This is substantial partial progress, but it does not settle the full separating-degenerating-slopes conjecture.

Current status (as of October 2026): The reducible–reducible case is claimed as proved, while the full conjecture on separating degenerating slopes remains open.

Qiu–Zhang conjecture on separating degenerating slopes

Primary source

Additional references

Progress summary

Solutions 0