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Transactions of the American Mathematical Society

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Algebra for Heckoid groups


Author: Robert Riley
Journal: Trans. Amer. Math. Soc. 334 (1992), 389-409
MSC: Primary 57M25
DOI: https://doi.org/10.1090/S0002-9947-1992-1107029-9
MathSciNet review: 1107029
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Abstract: We introduce an infinite collection of (Laurent) polynomials associated with a $ 2$-bridge knot or link normal form $ K = (\alpha ,\beta )$. Experimental evidence suggests that these "Heckoid polynomials" define the affine representation variety of certain groups, the Heckoid groups, for $ K$ . We discuss relations which hold in the image of the generic representation for each polynomial. We show that, with a certain change of variable, each Heckoid polynomial divides the nonabelian representation polynomial of $ L$ , where $ L$ belongs to an infinite collection of $ 2$-bridge knots/links determined by $ K$ and the Heckoid polynomial. Finally, we introduce a "precusp polynomial" for each $ 2$-bridge knot normal form, and show it is the product of two (possibly reducible) non-constant polynomials. We are preparing a sequel on the Heckoid groups and the evidence for some of the geometrical assertions stated in the introduction.


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DOI: https://doi.org/10.1090/S0002-9947-1992-1107029-9
Article copyright: © Copyright 1992 American Mathematical Society

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