Homomorphisms of knot groups on finite groups

Riley, Robert

doi:10.1090/S0025-5718-1971-0295332-4

Homomorphisms of knot groups on finite groups
HTML articles powered by AMS MathViewer

by Robert Riley PDF

Math. Comp. 25 (1971), 603-619 Request permission

Abstract:

We describe trial and error computer programs for finding certain homomorphisms of a knot group on a special projective group $LF(2,p),p$ prime, and programs to evaluate ${H_1}(\mathfrak {M};{\text {Z}})$ where $\mathfrak {M}$ is a finitely sheeted branched covering space of ${S^3}$ associated with such a homomorphism. These programs have been applied to several collections of examples, in particular to the Kinoshita-Terasaka knots, and we state numerous conjectures based on these experiments.

References

W. Burnside, Theory of groups of finite order, Dover Publications, Inc., New York, 1955. 2d ed. MR 0069818
J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) Pergamon, Oxford, 1970, pp. 329–358. MR 0258014
H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 14, Springer-Verlag, Berlin-Göttingen-New York, 1965. MR 0174618
Richard H. Crowell and Ralph H. Fox, Introduction to knot theory, Ginn and Company, Boston, Mass., 1963. Based upon lectures given at Haverford College under the Philips Lecture Program. MR 0146828
R. H. Fox, On the complementary domains of a certain pair of inequivalent knots, Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indagationes Math. 14 (1952), 37–40. MR 0048024
R. H. Fox, A quick trip through knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 120–167. MR 0140099
R. H. Fox, Construction of simply connected $3$-manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 213–216. MR 0140116
Fujitsugu Hosokawa, On $\nabla$-polynomials of links, Osaka Math. J. 10 (1958), 273–282. MR 102820
B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
Shin’ichi Kinoshita and Hidetaka Terasaka, On unions of knots, Osaka Math. J. 9 (1957), 131–153. MR 98386
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
Wilhelm Magnus and Ada Peluso, On knot groups, Comm. Pure Appl. Math. 20 (1967), 749–770. MR 222880, DOI 10.1002/cpa.3160200407

Knotentheorie

Horst Schubert, Über eine numerische Knoteninvariante, Math. Z. 61 (1954), 245–288 (German). MR 72483, DOI 10.1007/BF01181346
Horst Schubert, Knoten mit zwei Brücken, Math. Z. 65 (1956), 133–170 (German). MR 82104, DOI 10.1007/BF01473875