Hostname: page-component-f554764f5-nt87m Total loading time: 0 Render date: 2025-04-11T02:34:19.025Z Has data issue: false hasContentIssue false
  • English
  • Français

The Residual Finiteness of the Classical Knot Groups

Published online by Cambridge University Press:  20 November 2018

E. J. Mayland Jr.
E. J. Mayland Jr.*
Affiliation:
York University, Downsview, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of this paper is to extend the class of knot groups whose commutator subgroups are known to be residually a finite pgroup (i.e., residually of order a power of the prime p). Such a knot group is known to be residually finite (see, e.g., [10]), and although this class is quite restricted we will show that it includes all the groups of knots in the classical knot table [15].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 5 , 01 October 1975 , pp. 1092 - 1099
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Allenby, R. and Gregorac, R., Residual properties of nilpotent and super solvable groups, J. Algebra 23 (1972), 565–573.Google Scholar
2. Baumslag, G., On the residual finiteness of generalized free products of nilpotent groups, Trans. Amer. Math. Soc. 106 (1963), 193–209.Google Scholar
3. Baumslag, G., Groups with the same lower central sequence as a relatively free group, I. The groups, Trans. Amer. Math. Soc. 129 (1967), 308–321.Google Scholar
4. Baumslag, G., Groups with the same lower central sequence as a relatively free group, II. Properties, Trans. Amer. Math. Soc. Ufi (1969), 507–538.Google Scholar
5. Crowell, R. H. and Trotter, H., A class of pretzel knots, Duke Math. J. 30 (1963), 373–377.Google Scholar
6. Gruenberg, K. W., Residual properties of infinite soluble groups, Proc. London Math. Soc. 7 (1957), 29–62.Google Scholar
7. Iwasawa, K., Einige Sàtze uber freier gruppen, Proc. Japan Acad. 19 (1963), 272–274.Google Scholar
8. Karrass, A. and Solitar, D., The subgroups of a free product of two groups with an amalgamated subgroup, Trans. Amer. Math. Soc. 150 (1970), 227–255.Google Scholar
9. Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory: presentations of groups in terms of generators and relations (Pure and Applied Math., Vol. 13, Interscience, N.Y., 1966).Google Scholar
10. Mayland, E. J. Jr., On residually finite knot groups, Trans. Amer. Math. Soc. 168 (1972), 221–232.Google Scholar
11. Mayland, E. J., Two-bridge knots have residually finite groups (Proceedings Second International Conference on Group Theory, Canberra, 1973, Springer, 1974, 488–493).Google Scholar
12. Mayland, E. J., The residual finiteness of the groups of classical knots (to appear: Proceedings Geometric Topology Conference, Park City, Utah, 1974, Springer, 1975).Google Scholar
13. Neumann, H., Generalized free products with amalgamated subgroups. II, Amer. J. Math. 71 (1949), 491–540.Google Scholar
14. Neuwirth, L., Knot groups (Princeton Univ. Press, Princeton, N.J. 1965).Google Scholar
15. Reidemeister, K., Knotentheorie (Chelsea, New York, 1948).Google Scholar
16. Seifert, H., Uber das geschlecht von knoten, Math. Ann. 110 (1935), 571–592.Google Scholar
17. Trotter, H., Homology of group systems with applications to knot theory, Ann. Math. 76 (1963), 464–498.Google Scholar