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St. Petersburg Mathematical Journal

This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.

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

The 2020 MCQ for St. Petersburg Mathematical Journal is 0.68.

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Twist number of (closed) braids
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by A. V. Malyutin
Translated by: N. Yu. Netsvetaev
St. Petersburg Math. J. 16 (2005), 791-813
DOI: https://doi.org/10.1090/S1061-0022-05-00879-4
Published electronically: September 21, 2005
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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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Bibliographic 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
  • 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).
  • © Copyright 2005 American Mathematical Society
  • Journal: St. Petersburg Math. J. 16 (2005), 791-813
  • MSC (2000): Primary 57M25, 57M50
  • DOI: https://doi.org/10.1090/S1061-0022-05-00879-4
  • MathSciNet review: 2106667