The Weak Generalized Property R Conjecture

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An R-link is a link LL such that 0-surgery on LL yields a connected sum of copies of S1×S2S^1\times S^2. A 0-framed unlink is an unlink whose components all have framing zero, and a split collection of Hopf pairs is a collection of split Hopf-link pairs. The Weak Generalized Property R Conjecture. For every R-link LL, there is a 0-framed unlink UU and a split collection of Hopf pairs VV such that the split union L⊔U⊔VL\sqcup U\sqcup V is handleslide-equivalent to an unlink and a split collection of Hopf pairs.

This is a weaker variant of the Generalized Property R Conjecture. The source says that the GST links satisfy the weak conjecture, so this candidate is solved.

References

Primary source

Wenjie Diao, Haoqian Pan and Chunxing Yan, “Some experimental results on stable equivalence of GST Links for the Generalized Property R Conjecture”, arXiv:2604.17737 (2026).

Additional references

5 papers in this index state this conjecture (2009–2026). The statement above is taken from the most recent of them; the others are arXiv:2603.05664, arXiv:1507.06561, arXiv:1103.1601, arXiv:0908.2795.

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