On residually finite knot groups

Mayland, E. J.

doi:10.1090/S0002-9947-1972-0295329-1

On residually finite knot groups

Author: E. J. Mayland
Journal: Trans. Amer. Math. Soc. 168 (1972), 221-232
MSC: Primary 55A25
DOI: https://doi.org/10.1090/S0002-9947-1972-0295329-1
MathSciNet review: 0295329
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The residual finiteness of the class of groups of fibred knots, or those knot groups with finitely generated and, therefore, free commutator subgroups, has been known for some time. Using Baumslag’s results on absolutely parafree groups, this paper extends the result to twist knots (Whitehead doubles of the trivial knot) and certain other classes of nonfibred knots whose minimal spanning surface has complement with free fundamental group. As a by-product more explicit finite representations, namely cyclic extensions of certain $p$-groups, are obtained for these knot groups and the groups of fibred knots. Finally composites of two such knots also have residually finite groups.

Gilbert Baumslag, Groups with the same lower central sequence as a relatively free group. I. The groups, Trans. Amer. Math. Soc. 129 (1967), 308–321. MR 217157, DOI https://doi.org/10.1090/S0002-9947-1967-0217157-3
Gilbert Baumslag, Groups with the same lower central sequence as a relatively free group. II. Properties, Trans. Amer. Math. Soc. 142 (1969), 507–538. MR 245653, DOI https://doi.org/10.1090/S0002-9947-1969-0245653-3
D. B. A. Epstein, The degree of a map, Proc. London Math. Soc. (3) 16 (1966), 369–383. MR 192475, DOI https://doi.org/10.1112/plms/s3-16.1.369
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
K. W. Gruenberg, Residual properties of infinite soluble groups, Proc. London Math. Soc. (3) 7 (1957), 29–62. MR 87652, DOI https://doi.org/10.1112/plms/s3-7.1.29
Marshall Hall Jr., The theory of groups, The Macmillan Co., New York, N.Y., 1959. MR 0103215

A. G. Kuroš, Theory of groups, GITTL, Moscow, 1953; English transl., Vols. I, II, Chelsea, New York, 1956. MR 15, 501; MR 18, 188.

Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1966. MR 0207802

D. R. McMillan, Jr., Boundary preserving mappings of $3$-manifolds, Topology of Manifolds, Markham, Chicago, Ill., 1969.

B. H. Neumann, An essay on free products of groups with amalgamations, Philos. Trans. Roy. Soc. London Ser. A 246 (1954), 503–554. MR 62741, DOI https://doi.org/10.1098/rsta.1954.0007
Hanna Neumann, Generalized free products with amalgamated subgroups, Amer. J. Math. 70 (1948), 590–625. MR 26997, DOI https://doi.org/10.2307/2372201
Hanna Neumann, Generalized free products with amalgamated subgroups. II, Amer. J. Math. 71 (1949), 491–540. MR 30522, DOI https://doi.org/10.2307/2372346

L. Neuwirth, Knot theory, Princeton Univ. Press, Princeton, N.J., 1965.

Horst Schubert, Die eindeutige Zerlegbarkeit eines Knotens in Primknoten, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (1949), no. 3, 57–104 (German). MR 0031733

H. Seifert and W. Threlfall, Lehrbuch der Topologie, Chelsea, New York, 1947.

Peter Stebe, Residual finiteness of a class of knot groups, Comm. Pure Appl. Math. 21 (1968), 563–583. MR 237621, DOI https://doi.org/10.1002/cpa.3160210605
Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI https://doi.org/10.2307/1970594

J. H. C. Whitehead, On doubled knots, J. London Math. Soc. 12 (1937), 63-71.