Pseudocharacters of braid groups and prime links

Malyutin, A.

doi:10.1090/S1061-0022-10-01093-9

Pseudocharacters of braid groups and prime links
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by A. V. Malyutin
Translated by: the author

St. Petersburg Math. J. 21 (2010), 245-259

DOI: https://doi.org/10.1090/S1061-0022-10-01093-9

Published electronically: January 21, 2010

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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 $|\phi (\beta )|>C_{\phi }$, 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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