|
Dehornoy's ordering on the braid group and braid moves
Author(s):
A.
V.
Malyutin;
N.
Yu.
Netsvetaev
Translated by:
the authors
Original publication:
Algebra i Analiz,
tom 15
(2003),
vypusk 3.
Journal:
St. Petersburg Math. J.
15
(2004),
437-448.
MSC (2000):
Primary 57M25
Posted:
March 30, 2004
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
In terms of Dehornoy's ordering on the braid group  , 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  that are not minimal lie in a finite interval of Dehornoy's ordering.
References:
-
- 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
Email:
malyutin@pdmi.ras.ru
N.
Yu.
Netsvetaev
Affiliation:
St. Petersburg State University, Faculty of Mathematics and Mechanics, Universitetskii pr. 28, Petrodvorets, St. Petersburg 198504, Russia
Email:
nn@pdmi.ras.ru
DOI:
10.1090/S1061-0022-04-00816-7
PII:
S 1061-0022(04)00816-7
Received by editor(s):
23/NOV/2002
Posted:
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).
Copyright of article:
Copyright
2004,
American Mathematical Society
|
Comments: webmaster@ams.org