Conjecture 12.4 — (3-move conjecture, Y.

Definitions and mathematical statement

In the usual unframed link setting, a link is a finite disjoint union of embedded circles in $S^3$, up to ambient isotopy. A trivial link is an unlink: its components bound pairwise disjoint embedded disks. A standard $3$-move replaces inside a ball a trivial two-strand tangle by three consecutive half-twists, or performs the inverse replacement, leaving the link outside the ball fixed. The precise local picture and orientation convention must agree with the cited source.

Conjecture 12.4, the $3$-move conjecture attributed to Y. Nakanishi [305], asserts: \[ \forall\text{ links }L\ \exists\text{ an unlink }U:\quad L\text{ is related to }U\text{ by finitely many }3\text{-moves and isotopies}. \] The statement does not require the unlink to have a prescribed number of components. The supplied research assessment reports that this universal assertion is false: there exists a link that cannot be related by such a sequence to any trivial link. It attributes the negative resolution to Dabkowski--Przytycki (2002), using a Burnside-group obstruction. This is a source-reported resolution; no counterexample or proof is reconstructed here. The generic background note calling the source item open is retained as conflicting status evidence, not used to overwrite the explicit negative assessment.

Short English statement

The conjecture claimed that every link becomes an unlink through finitely many $3$-moves. The supplied registry marks it solved negatively and cites a 2002 Burnside-group obstruction. This draft records the original universal claim and its reported disproof, with the local-move convention still requiring source review.

Sources

• Local UnsolvedMath registry, permanent ID $10400210$, exact source code catalogue entry, Conjecture 12.4, attributed to Y. Nakanishi [305]. The bibliography entry [305] is absent; the title is truncated.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Conjecture 12.4, PDF page 160: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Bold headings distinguish 231 numbered items from surrounding references.
• Supplied disproof citation: M. K. Dabkowski and J. H. Przytycki, "Burnside obstructions to the Montesinos-Nakanishi 3-move conjecture," Geom. Topol. 6 (2002), 355--360, arXiv:math/0205057. The research summary describes a Burnside-group obstruction.
• Supplied further references: Dabkowski--Przytycki, "Unexpected links between components of the same link," Fund. Math. (2004), for further counterexamples; an earlier Przytycki note (1999) on the $(3,2)$-move conjecture, without full details. The summary cross-refers to Problem 4.10, source code AMR-103-0092, reporting disproof in February 2002.
• Extraction pdf-font-structure, accessed 2026-07-29. Registry status solved and classification SOLVED-IN-LITERATURE report a negative resolution, while generic source evidence NEEDS_REVIEW calls the item open and asks for a current primary-literature audit.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.