Volume 4 (2002)

Download this article
For screen
For printing
Recent Volumes
Volume 1, 1998
Volume 2, 1999
Volume 3, 2000
Volume 4, 2002
Volume 5, 2002
Volume 6, 2003
Volume 7, 2004
Volume 8, 2006
Volume 9, 2006
Volume 10, 2007
Volume 11, 2007
Volume 12, 2007
Volume 13, 2008
Volume 14, 2008
Volume 15, 2008
Volume 16, 2009
Volume 17, 2011
Volume 18, 2012
Volume 19, 2015
The Series
MSP Books and Monographs
About this Series
Editorial Board
Ethics Statement
Author Index
Submission Guidelines
Author Copyright Form
ISSN (electronic): 1464-8997
ISSN (print): 1464-8989
Other MSP Publications

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


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.


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


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

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