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Hecke invariants of knot groups

Published online by Cambridge University Press:  18 May 2009

Robert Riley
Robert Riley
Affiliation:
University of Southampton
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For each characteristic p, let Fp be the prime field and let Ώp be a fixed universal field which is algebraically closed and of infinite transcendence degree over Fp. When p = 0 we take Ώp = ℂ. Let F be a subfield of Ώp and let R be an integral domain whose quotient field is F. We abbreviate SL(2, R), PGL(2, R), PSL(2, R) to SL(R), PGL(R), PSL(R) respectively, and we cohsider PSL(R) as a group of projective transformations of the projective line ℘(Ώp) and of the “subline” ℘(F) ⊂ ℘(ΏP). The elements of PSL(R) are classified by the number of fixed points they have on ℘(F). If x ∈ PSL(R) has one such fixed point P, then P is the unique fixed point of x on ℘(ΏP) and x is called parabolic. All other x (except the identity E) have two distinct fixed points on ℘(Ώp) and x is called hyperbolic if these are on ℘(F), and elliptic otherwise. We put symbols for operators on the right.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 15 , Issue 1 , March 1974 , pp. 17 - 26
Copyright
Copyright © Glasgow Mathematical Journal Trust 1974

References

REFERENCES

1.Fox, R. H., A quick trip through knot theory, Topology of 3-manifolds and related topics (Englewood Cliffs, N.J., 1962), 120–167.Google Scholar
2.Fox, R. H., Metacyclic invariants of knots and lines, Canad. J. Math. 22 (1970), 193–201.CrossRefGoogle Scholar
3.Fricke, R., Lehrbuch der Algebra, Dritter Band (Braunschweig, 1928).Google Scholar
4.Lehner, J., A short course in automorphic functions (New York, 1966).Google Scholar
5.Maskit, B., On Poincaré's theorem for fundamental polygons, Advances in Mathematics 7 (1971), 219–230.CrossRefGoogle Scholar
6.Riley, R., Homomorphisms of knot groups on finite groups, Math. Comp. 25 (1971), 603–619.CrossRefGoogle Scholar
7.Riley, R., Parabolic representations of knot groups, I, Proc. London Math. Soc. (3) 24 (1972), 217–242.CrossRefGoogle Scholar
8.Singerman, D., Subgroups of Fuchsian groups and finite permutation groups, Bull. London Math. Soc. 2 (1970), 319–323.CrossRefGoogle Scholar
9.Evans, R., A fundamental region for Hecke's modular group, J. Number Theory 5 (1973), 108–115.CrossRefGoogle Scholar