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Burnside groups in knot theory
by
Mieczyslaw K. Dabkowski
University of Texas at Dallas
Coauthors: Jozef H. Przytycki (GWU)
In classical Knot Theory a substantial amount of effort has been devoted to the search for unknotting moves on links. Tangle moves which unknot or at least simplify significantly every link play an important role in Knot Theory and 3-dimensional topology. Good understanding of these moves allows us to analyze skein modules obtained by their deformations. For example, a deformation of a crossing change leads to the Jones polynomial of links and more generally to the Homflypt skein module relation. We answer the question whether rational [p/q]-moves (p an odd prime) can simplify any link into a trivial one. We show that pth Burnside group of a link can be an obstruction to the simplification. In particular we disprove Harikae-Nakanishi (2, 2)-move conjecture, showing that the knots 940 and 949 cannot be trivialized. We also answers Kawauchi's question, showing that half" 2-cabling of the Whitehead link cannot be reduced into a trivial ink via 4-moves.
Paper reference: arXiv:math.GT/0309140
Date received: May 27, 2004
Copyright © 2004 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 # canv-30.