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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Twist number of (closed) braids


Author: A. V. Malyutin
Translated by: N. Yu. Netsvetaev
Original publication: Algebra i Analiz, tom 16 (2004), nomer 5.
Journal: St. Petersburg Math. J. 16 (2005), 791-813
MSC (2000): Primary 57M25, 57M50
Published electronically: September 21, 2005
MathSciNet review: 2106667
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Abstract | References | Similar Articles | Additional Information

Abstract: A real-valued invariant of (closed) braids, called the twist number, is introduced and studied. This invariant is effectively computable and has clear geometric sense.

As a functional on the braid group, the twist number is a pseudocharacter (i.e., a function that is ``almost'' a homomorphism). It is closely related to Dehornoy's ordering (and to all Thurston-type orderings) on the braid group. In special cases, the twist number coincides with some characteristics introduced by William Menasco.

In terms of the twist number, restrictions are established on the applicability of the Markov destabilization and Birman-Menasco moves on closed braids. These restrictions were conjectured by Menasco (Kirby's problem book, 1997). As a consequence, conditions for primality of the link represented by a braid are obtained.

The results were partially announced in an earlier paper.


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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

DOI: http://dx.doi.org/10.1090/S1061-0022-05-00879-4
PII: S 1061-0022(05)00879-4
Keywords: Braid group, Markov destabilization, Birman--Menasco moves, link theory
Received by editor(s): January 31, 2003
Published electronically: September 21, 2005
Additional Notes: This work was partly supported by grant PD02-1.1-423 from Russian Ministry of Education and by grant NSh–1914.2003.1 from President of Russia (Support of Leading Scientific Schools).
Article copyright: © Copyright 2005 American Mathematical Society