Nakanishi's 4-move conjecture for knots
A 4-move is the local operation on a link diagram that replaces four consecutive half-twists between two strands by the opposite four half-twists. Nakanishi's 4-move conjecture. Every knot can be reduced to the trivial knot by 4-moves. This conjecture is an early move-equivalence conjecture in knot theory, formulated by Nakanishi in 1979 before the 3-move conjecture was formally stated; the supplied evidence says it is disproved by the -cable of the figure-eight knot with a half-twist, a knot with 17 crossings.
References
Primary source
Rhea Palak Bakshi, Benjamin A. Burton, Huizheng Guo, Dionne Ibarra, Gabriel Montoya-Vega, Sujoy Mukherjee and Józef H. Przytycki, “The Montesinos-Nakanishi 3-move conjecture for links up to 20 crossings”, arXiv:2502.17711 (2025).
Additional references
5 papers in this index state this conjecture (2001–2025). The statement above is taken from the most recent of them; the others are arXiv:math/0601004, arXiv:math/0312527, arXiv:math/0303012, arXiv:math/0109029.
Progress summary
A 2025 paper reports a -crossing counterexample, but that disproof has not been independently verified and the conjecture remains formally unsettled.
Nakanishi formulated the conjecture in 1979: every knot should be reducible to the trivial knot by -moves.
Known results
- Verified for -bridge knots, closed -braid knots, and all knots with at most crossings (2012 and 2020).
- Extensive computations found no counterexample among most alternating knots through crossings (2012).
- The Burnside-group method gives no obstruction to the knot conjecture (2003).
February 2025 candidate counterexample; 2026 weaker theorem
A February 2025 paper reports that the -cable of the figure-eight knot with a half-twist, a -crossing knot, is a guessed counterexample; this is an unverified claim rather than a proved disproof. A 2026 paper proves only that every knot is -move equivalent to one with trivial Alexander polynomial, and explicitly leaves Nakanishi’s conjecture open.
Current status (as of September 2026): The conjecture has a reported but unverified -crossing counterexample; no verified disproof or proof is recorded, and the 2026 weaker theorem does not settle it.
Sources
- ar5iv.labs.arxiv.org
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Solutions 0
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