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5-MOVE EQUIVALENCE CLASSES OF LINKS AND THEIR ALGEBRAIC INVARIANTS

MIECZYSŁAW K. DABKOWSKI

Department of Mathematics, University of Texas at Dallas, Richardson, TX 75083, USA

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MAKIKO ISHIWATA

Osaka City University, OCAMI, Osaka-City 558-8585, Japan

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, and

JÓZEF H. PRZYTYCKI

Department of Mathematics, George Washington University, USA

On sabbatical leave at UMD, College Park, USA.

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https://doi.org/10.1142/S0218216507005877Cited by:2 (Source: Crossref)

Abstract

We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main tools are Jones and Kauffman polynomials and the fundamental group of the 2-fold branch cover of S³ along a link. We use also the fact that a 5-move is a composition of two rational ±(2, 2)-moves (i.e. -moves) and rational moves can be analyzed using the group of Fox colorings and its non-abelian version, the Burnside group of a link. One curious observation is that links related by one (2, 2)-move are not 5-move equivalent. In particular, we partially classify (up to 5-moves) 3-braids, pretzel and Montesinos links, and links up to 9 crossings.

Dedicated to Lou Kauffman on His 60th Birthday

Keywords:

AMSC: 57M25, 57M27

References

D. Bar-Natan, http://www.math.toronto.edu/drorbn/KAtlas/Links . Google Scholar
R. D. Brandt, W. B. R. Lickorish and K. C. Millett, Invent. Math. 84, 563 (1986), DOI: 10.1007/BF01388747. Crossref, Web of Science, Google Scholar
J. H. Conway, An enumeration of knots and links and some of their algebraic properties, Proceedings of the Conference on Computational Problems in Abstract Algebra held at Oxford in 1967, ed. J. Leech (Pergamon Press, 1970) pp. 329–358. Google Scholar
H. S. M. Coxeter, Factor groups of the braid group, Proc. Fourth Canadian Math. Congress, Banff (1957) pp. 95–122. Google Scholar
M. K. Dabkowski, M. Ishiwata and J. H. Przytycki, Rational moves and tangle embeddings: (2,2)-moves as a case study, Proceedings of the Conference Topology of Knot VII (held at TWCU, December 23–26) (2005) pp. 37–46. Google Scholar
M. K. Dabkowski and J. H. Przytycki, Geom. Topol. 6, 335 (2002), DOI: 10.2140/gt.2002.6.355. Google Scholar
M. K. Dabkowski and J. H. Przytycki, Proc. Natl. Acad. Science 101(50), 17357 (2004), DOI: 10.1073/pnas.0406098101. Crossref, Web of Science, Google Scholar
M. K. Dabkowski and J. H. Przytycki, Burnside groups in knot theory, preprint (2004) . Google Scholar
J. Dymara, T. Januszkiewicz and J. H. Przytycki, Symplectic structure on Colorings, Lagrangian tangles and Tits buildings, preprint (May 2001) . Google Scholar
R. Fenn and C. Rourke, J. Knot Theory Ramifications 1(4), 343 (1992), DOI: 10.1142/S0218216592000203. Link, Google Scholar
T. Harikae and Y. Uchida, Topics in Knot Theory, NATO Advanced Science Insitutes Series C 399, ed. M. Bozhüyük (Kluwer Academic Publisher, 1993) pp. 269–276. Crossref, Google Scholar
C. F. Ho, Abstracts Amer. Math. Sci. 6(4), 300 (1985). Google Scholar
J. Hoste and J. H. Przytycki, A survey of skein modules of 3-manifolds, Knots 90, Proceedings of the International Conference on Knot Theory and Related Topics, Osaka (Japan), August 15–19, 1990, ed. A. Kawauchi (Walter de Gruyter, 1992) pp. 363–379. Google Scholar
S. Jablan and R. Sazdanovic , LinKnot; Knot Theory by Computer , Series on Knots and Everything 21 ( World Scientific Publishing Co. , 2007 ) . Crossref, Google Scholar
V. F. R. Jones, Ann. Math. 126(2), 335 (1987). Crossref, Web of Science, Google Scholar
D. Joyce, J. Pure Appl. Alg. 23, 37 (1982), DOI: 10.1016/0022-4049(82)90077-9. Crossref, Web of Science, Google Scholar
L. H. Kauffman, Trans. Amer. Math. Soc. 318(2), 417 (1990), DOI: 10.2307/2001315. Crossref, Web of Science, Google Scholar
L. H. Kauffman and S. Lambropoulou, Adv. Appl. Math. 33(2), 199 (2004), DOI: 10.1016/j.aam.2003.06.002. Crossref, Web of Science, Google Scholar
A. Kawauchi , A Survey of Knot Theory ( Birkhäusen Verlag , Basel, Boston, Berlin , 1996 ) . Google Scholar
R. Kirby, Problems in low-dimensional topology, Geometric Topology (Proceedings of the Georgia International Topology Conference, 1993)2, Studies in Advanced Mathematics, ed. W. Kazez (AMS/IP, 1997) pp. 35–473. Google Scholar
J. H. Przytycki, Braids, Contemporary Mathematics 78, eds. J. S. Birman and A. Libgober (1988) pp. 615–656. Crossref, Google Scholar
J. H. Przytycki, Bull. Ac. Pol. Math. 39(1–2), 91 (1991). Web of Science, Google Scholar
J. H. Przytycki, Knot Theory 42 (Banach Center Publications, 1998) pp. 275–295. Google Scholar
J. H. Przytycki , Three talks in Cuautitlan under the general title: Topologia algebraica basada sobre nudos , Proceedings of the First International Workshop on "Graphs — Operads — Logic, Cuautitlan, Mexico, March 12–16, 2001 . Google Scholar
J. H. Przytycki, Skein module deformations of elementary moves on links, Invariants of Knots and 3-Manifolds (Kyoto 2001)4, Geometry and Topology Monographs (2003) pp. 313–335. Google Scholar
J. H. Przytycki , From 3-moves to Lagrangian tangles and cubic skein modules , Advances in Topological Quantum Field Theory, Proceedings of the NATO ARW on New Techniques in Topological Quantum Field Theory, Kananaskis Village, Canada from 22to 26 August 2001 , ed. J. M. Bryden ( 2004 ) . Google Scholar
J. H. Przytycki, Algebraic topology based on knots: An introduction, Knots 96, Proceedings of the Fifth International Research Institute of MSJ, ed. S. Suzuki (World Scientific Publishing Co., 1997) pp. 279–297. Google Scholar
J. H. Przytycki, Kobe Math. J. 16(1), 45 (1999). Web of Science, Google Scholar
J. H. Przytycki and T. Tsukamoto, J. Knot Theory Ramifications 10(7), 959 (2001), DOI: 10.1142/S0218216501001281. Link, Web of Science, Google Scholar
D. Rolfsen , Knots and Links ( Publish or Perish , 1976 ) . Google Scholar
M. B. Thistlethwaite, Aspects of Topology, London Mathematics Society Lecture Notes Series 93 (1985) pp. 1–76. Crossref, Google Scholar