Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Dehornoy's ordering on the braid group and braid moves

Authors: A. V. Malyutin and N. Yu. Netsvetaev
Translated by: the authors
Original publication: Algebra i Analiz, tom 15 (2003), nomer 3.
Journal: St. Petersburg Math. J. 15 (2004), 437-448
MSC (2000): Primary 57M25
Published electronically: March 30, 2004
MathSciNet review: 2052167
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In terms of Dehornoy's ordering on the braid group ${\mathcal B}_n$, restrictions are found that prevent us from performing the Markov destabilization and the Birman-Menasco braid moves. As a consequence, a sufficient condition is obtained for the link represented by a braid to be prime, and it is shown that all braids in ${\mathcal B}_n$ that are not minimal lie in a finite interval of Dehornoy's ordering.

References [Enhancements On Off] (What's this?)

  • 1. J. S. Birman, Braids, links, and mapping class groups, Ann. of Math. Stud., vol. 82, Princeton Univ. Press, Princeton, NJ, 1974. MR 51:11477
  • 2. J. Birman and W. Menasco, Studying links via closed braids. IV. Composite links and split links, Invent. Math. 102 (1990), 115-139. MR 92g:57010a
  • 3. -, Studying links via closed braids. V. The unlink, Trans. Amer. Math. Soc. 329 (1992), 585-606. MR 92g:57010b
  • 4. -, Stabilization in the braid groups (with applications to transverse knots), Preprint, 2002.
  • 5. S. Burckel, The wellordering on positive braids, J. Pure Appl. Algebra 120 (1997), no. 1, 1-17. MR 98h:20062
  • 6. C. Cerf and A. Maes, A family of Brunnian links based on Edwards' construction of Venn diagrams, J. Knot Theory Ramifications 10 (2001), no. 1, 97-107. MR 2002i:57007
  • 7. P. Cromwell, Positive braids are visually prime, Proc. London Math. Soc. (3) 67 (1993), no. 2, 384-424. MR 95c:57008
  • 8. P. Dehornoy, Braids and self-distributivity, Progr. Math., vol. 192, Birkhäuser Verlag, Basel, 2000. MR 2001j:20057
  • 9. I. A. Dynnikov, Arc-presentations of links. Monotonic simplification, Preprint, 2002.
  • 10. R. Kirby (ed.), Problems in low-dimensional topology, Geometric Topology (Athens, GA, 1993), AMS/IP Stud. Adv. Math., vol. 2.2, Amer. Math. Soc., Providence, RI, 1997, pp. 35-473. MR 98f:57001
  • 11. R. Laver, Braid group actions on left distributive structures and well orderings in the braid groups, J. Pure Appl. Algebra 108 (1996), no. 1, 81-98. MR 97e:20061
  • 12. A. V. Malyutin, Orderings on braid groups, operations over closed braids, and confirmation of Menasco's conjecture, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 267 (2000), 163-169. (Russian) MR 2002a:57008
  • 13. -, Fast algorithms for the recognition and comparison of braids, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 279 (2001), 197-217. (Russian) MR 2002g:20061
  • 14. H. Short and B. Wiest, Orderings of mapping class groups after Thurston, Enseign. Math. (2) 46 (2000), 279-312. MR 2003b:57003

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 57M25

Retrieve articles in all journals with MSC (2000): 57M25

Additional Information

A. V. Malyutin
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

N. Yu. Netsvetaev
Affiliation: St. Petersburg State University, Faculty of Mathematics and Mechanics, Universitetskiĭ pr. 28, Petrodvorets, St. Petersburg 198504, Russia

Received by editor(s): November 23, 2002
Published electronically: March 30, 2004
Additional Notes: Partially supported by the RFBR (grant no. 01-01-01014) and the Russian Ministry of Education (grant PD02-1.1-423).
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society