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Pseudocharacters of braid groups and prime links


Author: A. V. Malyutin
Translated by: the author
Original publication: Algebra i Analiz, tom 21 (2009), nomer 2.
Journal: St. Petersburg Math. J. 21 (2010), 245-259
MSC (2000): Primary 20F36
DOI: https://doi.org/10.1090/S1061-0022-10-01093-9
Published electronically: January 21, 2010
MathSciNet review: 2549454
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Abstract: Pseudocharacters of groups have recently found an application in the theory of classical knots and links in  $ \mathbb{R}^3$. More precisely, there is a relationship between pseudocharacters of Artin's braid groups and the properties of links represented by braids. In the paper, this relationship is investigated and the notion of kernel pseudocharacters of braid groups is introduced. It is proved that if a kernel pseudocharacter $ \phi$ and a braid $ \beta$ satisfy $ \vert\phi(\beta)\vert>C_{\phi}$, where $ C_{\phi}$ is the defect of $ \phi$, then $ \beta$ represents a prime link (i.e., a link that is noncomposite, nonsplit, and nontrivial). Furthermore, the space of braid group pseudocharacters is studied and a way is described to obtain nontrivial kernel pseudocharacters from an arbitrary braid group pseudocharacter that is not a homomorphism. This makes it possible to employ an arbitrary nontrivial braid group pseudocharacter for the recognition of prime knots and links.


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Additional Information

A. V. Malyutin
Affiliation: St. Petersburg Branch, Steklov Institute of Mathematics, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: malyutin@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-10-01093-9
Keywords: Knot, link, braid, pseudocharacter, quasimorphism
Received by editor(s): September 16, 2008
Published electronically: January 21, 2010
Additional Notes: Supported in part by RFBR (grant 08-01-00379a) and the RF President Program for Support of leading Scientific Schools (grant NSh-2460.2008.1112)
Article copyright: © Copyright 2010 American Mathematical Society

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