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Burnside groups of knots, tangle moves and their skein module deformations
by
Jozef H. Przytycki
GWU and UMD
Coauthors: Mieczyslaw K. Dabkowski (UT Dallas)
We describe the use of Burnside groups in knot theory and topology of 3-dimensional manifolds. We define the nth Burnside group, Bn(G), of the given group G, as the quotient of G by a normal subgroup generated by all elements of G of the form wn. If G is a free group of k generates we obtain the classical (1902) Burnside group B(k, n). The nth Burnside group of a link L in S3 is defined to be Bn(L) = Bn(π1(ML(2))), where ML(2) is the double branch cover of S3 branched along L. Our method of Burnside group of links allows us to settle (disprove) conjectures of Montesinos-Nakanishi, Kawauchi, and Harikae-Nakanishi-Uchida about 3-moves, 4-moves, and (2, 2)-moves, respectively. We can also show that the manifold Mφ (suspected of being the smallest volume hyperbolic oriented 3-manifold) is not obtainable from (S1 ×S2)k by a finite sequence of q/5-surgeries.
Skein modules, on which most of research was concentrated till now,
are based on the deformation of a smoothing (1-move) and crossing change
(2-move). We argue that deformation of tangle moves (e.g. 3-moves, 4-moves,
(2, 2)-moves) can lead to new, powerful skein modules of 3-manifolds.
References:
http://arxiv.org/abs/math.GT/0501539
http://front.math.ucdavis.edu/math.GT/0205040
http://front.math.ucdavis.edu/math.GT/0309140
http://front.math.ucdavis.edu/math.GT/0405248
http://front.math.ucdavis.edu/math.GT/0109029
http://front.math.ucdavis.edu/math.GT/0010282
http://www.maths.warwick.ac.uk/gt/GTMon4/paper21.abs.html
Date received: April 11, 2007
Copyright © 2007 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 # cauq-14.