Algebra for Heckoid groups
HTML articles powered by AMS MathViewer
- by Robert Riley
- Trans. Amer. Math. Soc. 334 (1992), 389-409
- DOI: https://doi.org/10.1090/S0002-9947-1992-1107029-9
- PDF | Request permission
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.References
- Matthew A. Grayson, The orbit space of a Kleinian group: Riley’s modest example, Math. Comp. 40 (1983), no. 162, 633–646. MR 689478, DOI 10.1090/S0025-5718-1983-0689478-4
- Troels Jørgensen, Compact $3$-manifolds of constant negative curvature fibering over the circle, Ann. of Math. (2) 106 (1977), no. 1, 61–72. MR 450546, DOI 10.2307/1971158
- Tomotada Ohtsuki, Ideal points and incompressible surfaces in two-bridge knot complements, J. Math. Soc. Japan 46 (1994), no. 1, 51–87. MR 1248091, DOI 10.2969/jmsj/04610051
- Robert Riley, Parabolic representations of knot groups. I, Proc. London Math. Soc. (3) 24 (1972), 217–242. MR 300267, DOI 10.1112/plms/s3-24.2.217
- Robert Riley, Knots with the parabolic property $P$, Quart. J. Math. Oxford Ser. (2) 25 (1974), 273–283. MR 358758, DOI 10.1093/qmath/25.1.273
- Robert Riley, Applications of a computer implementation of Poincaré’s theorem on fundamental polyhedra, Math. Comp. 40 (1983), no. 162, 607–632. MR 689477, DOI 10.1090/S0025-5718-1983-0689477-2
- Robert Riley, Nonabelian representations of $2$-bridge knot groups, Quart. J. Math. Oxford Ser. (2) 35 (1984), no. 138, 191–208. MR 745421, DOI 10.1093/qmath/35.2.191
- Horst Schubert, Knoten mit zwei Brücken, Math. Z. 65 (1956), 133–170 (German). MR 82104, DOI 10.1007/BF01473875
- W. Thurston, Hyperbolic geometry and $3$-manifolds, Low-dimensional topology (Bangor, 1979) London Math. Soc. Lecture Note Ser., vol. 48, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 9–25. MR 662424
Bibliographic Information
- © Copyright 1992 American Mathematical Society
- 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