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Mat. Zametki, 2000, Volume 68, Issue 5, Pages 692–698 (Mi mz990)  

This article is cited in 2 scientific papers (total in 2 papers)

Strong Positivity in Right-Invariant Order on Braid Groups and Quasipositivity

S. Yu. Orevkov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Dehornoy constructed a right invariant order on the braid group $B_n$ uniquely defined by the condition $\beta_0\sigma_i\beta_1>1$, if $\beta_0,\beta_1$ are words in $\sigma_{i+1}^{\pm 1},…,\sigma_{n-1}^{\pm 1}$. A braid is called strongly positive if $\alpha\beta\alpha^{-1}>1$ for any $\alpha\in B_n$. In the present paper it is proved that the braid $\beta_0(\sigma_1\sigma_2…\sigma_{n-1})(\sigma_{n-1}\sigma_{n-2}…\sigma_1)$ is strongly positive if the word $\beta_0$ does not contain $\sigma_1^{\pm 1}$. We also provide a geometric proof of the result by Burckel and Laver that the standard generators of a braid group are strongly positive. Finally, we discuss relations between the right invariant order and quasipositivity.

DOI: https://doi.org/10.4213/mzm990

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English version:
Mathematical Notes, 2000, 68:5, 588–593

Bibliographic databases:

UDC: 515
Received: 30.12.1998

Citation: S. Yu. Orevkov, “Strong Positivity in Right-Invariant Order on Braid Groups and Quasipositivity”, Mat. Zametki, 68:5 (2000), 692–698; Math. Notes, 68:5 (2000), 588–593

Citation in format AMSBIB
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\jour Math. Notes
\yr 2000
\vol 68
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Orevkov S.Yu., “Some Examples of Real Algebraic and Real Pseudoholomorphic Curves”, Perspectives in Analysis, Geometry, and Topology: on the Occasion of the 60th Birthday of Oleg Viro, Progress in Mathematics, 296, ed. Itenberg I. Joricke B. Passare M., Birkhauser Verlag Ag, 2012, 355–387  crossref  mathscinet  zmath  isi  scopus  scopus
    2. Dehornoy P., “Laver'S Results and Low-Dimensional Topology”, Arch. Math. Log., 55:1-2, SI (2016), 49–83  crossref  mathscinet  zmath  isi  scopus  scopus
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