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Skein module deformations of elementary moves on links

Jozef H Przytycki

Geometry & Topology Monographs 4 (2002) 313–335

DOI: 10.2140/gtm.2002.4.313

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

Mathematical Subject Classification

Primary: 57M27

Secondary: 20D99

References
Publication

Received: 8 November 2002
Revised: 17 October 2003
Accepted: 1 November 2003
Published: 13 November 2003

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