Burnside Groups in Knot Theory

Yasutaka Nakanishi asked in 1981 whether a 3-move is an unknotting operation. In Kirby’s problem list, this question is called The Montesinos-Nakanishi 3-move conjecture. We define the nth Burnside group of a link and use the 3rd Burnside group to answer Nakanishi’s question. One of the oldest elementary formulated problems in classical Knot Theory is the 3move conjecture of Nakanishi. A 3-move on a link is a local change that involves replacing parallel lines by 3 half-twists (Fig. 1). Fig. 1 Conjecture 1 (Montesinos-Nakanishi, Kirby’s problem list; Problem 1.59(1), [Kir]) Any link can be reduced to a trivial link by a sequence of 3-moves. The conjecture has been proved to be valid for several classes of links: by Y.Nakanishi for links up to 10 crossings and Montesinos links, by J.Przytycki for links up to 11 crossings, Conway’s algebraic links and closed 3-braids, by Q.Chen for links up to 12 crossings, closed 4-braids and closed 5-braids with the exception of the class of γ̂ – the square of the center of the fifth braid group (γ = (σ1σ2σ3σ4) 10), by Przytycki and T.Tsukamoto for 3-algebraic links (including 3-bridge links), and Tsukamoto for (4, 5)-algebraic links (including 4-bridge links), [Kir, Chen, Pr-Tsu, Tsu]. Nakanishi presented in 1994, an example which he couldn’t reduce by 3-moves: the 2-parallel of the Borromean rings (L2BR). Remark 2 In [Pr-1] it was noted that 3-moves preserve the first homology of the double branched cover of a link L with Z3 coefficients (H1(M (2) L ;Z3)). Suppose that γ̂ (resp. L2BR) can be reduced by 3-moves to a trivial link. Since H1(M (2) γ̂ ;Z3) = Z 4 3 ,