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Burnside groups in Knot Theory: 1. Solution to the Montesinos-Nakanishi 3-move conjecture
by
Jozef H. Przytycki
George Washington University
Coauthors: Mieczyslaw Dabkowski (GWU)
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; i.e. to show that some links cannot be reduced to trivial links by 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 Gamma - the square of the center of the fifth braid group, by Przytycki and T.Tsukamoto for 3-algebraic links (including 3-bridge links), and Tsukamoto for (4, 5)-algebraic links (including 4-bridge links). Nakanishi presented in 1994, an example which he couldn't reduce by 3-moves: the 2-parallel of the Borromean rings (L2BR). We show that the links Gamma and L2BR are not 3-move reducible to trivial links.
Date received: May 15, 2002
Copyright © 2002 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cajk-02.