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.

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Jozef H Przytycki
Department of Mathematics, George Washington University
Washington, DC 20052, USA

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