Obstructions to the Montesinos-Nakanishi 3-move conjecture
by
Jozef H. Przytycki
George Washington University
Coauthors: Mieczys\law Dabkowski (GWU)

Yasutaka Nakanishi asked in 1981 whether a 3-move is an unknotting operation. This question is called, in the Kirby's problem list, the Montesinos-Nakanishi Conjecture. Various partial results have been obtained by Q.Chen, Y.Nakanishi, J.Przytycki and T.Tsukamoto. Nakanishi and Chen presented examples which they couldn't reduce (the Borromean rings and the closure of the square of the center of the fifth braid group, [^g], respectively). The only tool, to analyze 3-move equivalence, till 1999, was the Fox 3-coloring (the number of Fox 3-colorings is unchanged by a 3-move). It allowed to distinguish different trivial links but didn't separate Nakanishi and Chen examples from trivial links. The group of tricolorings of a link L corresponds to the first homology group with Z₃ coefficients of the double branched cover of a link L, M_L⁽²⁾, i.e.

Tri(L) = H1(ML(2), Z3) + Z3

We find more delicate invariants of 3-moves using homotopy in place homology and we consider the fundamental group of M_L⁽²⁾. We define an nth Burnside group of a link as the quotient of the fundamental group of the double branched cover of the link divided by all relations of the form aⁿ=1. For n=2, 3, 4, 6 the quotient group is finite.

The third Burnside group of a link is unchanged by 3-moves.

In the proof we use the "core" presentation of the group from the diagram; that is arcs are generators and each crossing gives a relation c=ab^-1a where a corresponds to the overcrossing and b and c to undercrossings.

The Montesinos-Nakanishi 3-move conjecture does not hold for Chen's example [^g].

To show that [^g] has different third Burnside group than any trivial link it suffices to show that the following element, P, of the Burnside free group B(4, 3)={x, y, z, t: (a)³} is nontrivial:
P=uwtu^-1w^-1t^-1 where

u=xy^-1zt^-1 and w=x^-1yz^-1t. With the help of GAP it has been achieved!! (Feb. 22, 2002).

We have confirmed our calculation using Magnus program.

Date received: March 15, 2002