Braid Groups

Jahrme Risner

Why Study Braids?

    Where did they come from?

    Emil Artin

    1925

    Artinian Braid Group


    Joan Birman

    1975

    What is a Braid?

    A braid is a representation of the movements of points.

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    Composition of Braids in \(\mathcal{B}_3\)

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    Inverse and Identity in \(\mathcal{B}_2\)

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    The Pure Braid Groups

    Pure Braid: Braid inducing trivial permutation.

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    Pure Braid Group: Group of pure braids on \(n\)-strands.

    Center of the Braid Group

    Center of \(\mathcal{B}_n\) (\(n \gt 2\)) is \(\langle \Delta^2 \rangle\) where \(\Delta = \sigma_1 (\sigma_2\sigma_1)(\sigma_3\sigma_2\sigma_1)\cdots(\sigma_{n-1}\sigma_{n-2}\cdots\sigma_1).\)

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    References

    [1]
      
    Emil Artin, Theory of Braids, Annals of Mathematics (1947).
    [2]
      
    Emil Artin, The Theory of Braids, American Scientist 38 no. 1 (1950).
    [3]
      
    Joan Birman, Braids, Links, and Mapping Class Groups, (1975).
    [4]
      
    Joan Birman and Tara Brendle, Braids: A Survey, Handbook of Knot Theory (2005).
    [5]
      
    David Garber, Braid Group Cryptography, Braids: Introductory Lectures on Braids, Configurations and Their Applications no. 19 (2010).
    [6]
      
    Daniel Glasscock, What is a Braid Group?, (2012).
    [7]
      
    Nicholas Jackson, Notes on Braid Groups, (2004).
    [8]
      
    Joshua Lieber, Introduction to Braid Groups, (2011).
    [9]
      
    Juan Gonzalez-Meneses, Basic Results on Braid Groups, Annales Mathematiques Blaise Pascal 18 no. 1 (2011).

    Q&A

    Copyright

    ©2016 Jahrme Risner GFDL License
    Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the appendix entitled “GNU Free Documentation License.”