Geometry & Topology Monographs, Vol. 4 (2002),
Invariants of knots and 3-manifolds (Kyoto 2001),
Paper no. 21, pages 313--335.
Skein module deformations of elementary moves on links
Jozef H Przytycki
Abstract.
This paper is based on my talks (`Skein modules with a cubic skein
relation: properties and speculations' and `Symplectic structure on
colorings, Lagrangian tangles and its applications') given in Kyoto
(RIMS), September 11 and September 18 respectively, 2001. The first
three sections closely follow the talks: starting from elementary
moves on links and ending on applications to unknotting number
motivated by a skein module deformation of a 3-move. The theory of
skein modules is outlined in the problem section of these
proceedings.
In the first section we make the point that despite its long history,
knot theory has many elementary problems that are still open. We
discuss several of them starting from the Nakanishi's 4-move
conjecture. In the second section we introduce the idea of Lagrangian
tangles and we show how to apply them to elementary moves and to
rotors. In the third section we apply (2,2)-moves and a skein module
deformation of a 3-move to approximate unknotting numbers of knots. In
the fourth section we introduce the Burnside groups of links and use
these invariants to resolve several problems stated in section 1.
Keywords.
Knot, link, skein module, $n$-move, rational move, algebraic tangle, Lagrangian tangle, rotor, unknotting number, Fox coloring, Burnside group, branched cover
AMS subject classification.
Primary: 57M27.
Secondary: 20D99.
E-print: arXiv:math.GT/0312527
Submitted to GT on 8 November 2002.
(Revised 17 October 2003.)
Paper accepted 1 November 2003.
Paper published 13 November 2003.
Notes on file formats
Jozef H Przytycki
Department of Mathematics, George Washington University
Washington, DC 20052, USA
Email: przytyck@gwu.edu
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