Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T00:47:35.640Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Structural Property of the Groups of Alternating Links

Published online by Cambridge University Press:  20 November 2018

E. J. Mayland Jr  and
Kunio Murasugi
E. J. Mayland Jr
Affiliation:
York University, Downsview, Ontario
Kunio Murasugi
Affiliation:
University of Toronto, Toronto, 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.

In this paper, we will prove, as a consequence of the main theorem,

THEOREM A. (See Corollary 2.6). The group of an alternating knot, for which the leading coefficient of the knot polynomial is a prime power, is residually finite and solvable.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 3 , 01 June 1976 , pp. 568 - 588
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Bankwitz, C., Uber die Torsionzahlen der alternierenden Knoten, Math. Ann. 103 (1930), 145–161.Google Scholar
2. Baumslag, G., Groups with the same lower central sequence as a relatively free group. I: The groups. II: Properties, Trans. Amer. Math. Soc. 129 (1967), 308-321, and lift (1969), 507–538.Google Scholar
3. Boler, J. and Evans, B., The free product of residually finite groups amalgamated along retracts is residually finite, Proc. Amer. Math. Soc. 37 (1973), 50–52.Google Scholar
4. Brauner, K., Zur Géométrie der Functionen zweier komplexer Veranderlicher, Abh. Math. Sem. Hamburg Univ. 9 (1928), 1–55.Google Scholar
5. Brown, E. and Crowell, R., The augmentation subgroup of a link, J. Math, and Mech., IS (1966), 1065–1074.Google Scholar
6. Crowell, R., The genus of alternating link types, Ann. Math. (1959), 259–275.Google Scholar
7. Fox, R. H., Some problems in knot theory, Topology of 3-manifolds and related topics, Proc. The Univ. of Georgia Institute, 1961, pp. 168–176 (Prentice-Hall, Englew∞d Cliffs, N.J., 1962).Google Scholar
8. Gruenberg, K. W., Residual properties of infinite soluble groups, Proc. London Math. Soc. (3) 7 (1957), 29–62.Google Scholar
9. Hashizume, Y., On the uniqueness of the decomposition of a link, Osaka Math. J. 10 (1966), 1065–1074.Google Scholar
10. Iwasawa, K., Einige Sàtze uber fréter Gruppen, Proc. Japan Acad. 19 (1963), 272–274.Google Scholar
11. Magnus, W., Karrass, A., and Solitar, D., Combinatorial group theory: Presentations of groups in terms of generators and relations, Pure and Appl. Math., vol. 13 (Interscience, N.Y., 1966).Google Scholar
12. Mayland, E. J., Jr., On residually finite knot groups, Trans. Amer. Math. Soc. 168 (1972), 221–232.Google Scholar
13. Mayland, E. J., Jr., Two-bridge knots have residually finite groups, Proc. Second International Conference on the theory of groups, Newman, M. F., edt., pp. 488–493 (Lecture notes, No. 372, Springer, Berlin, 1974).Google Scholar
14. Newman, M. F., The residual finitene s s of the classical knot groups, Can. J. Math. 17 (1975), 1092- 1099.Google Scholar
15. Milnor, J., Singular points of complex hypersurfaces (Princeton U. Press, Princeton, N.J., 1968).Google Scholar
16. Murasugi, K., On the genus of the alternating knot, I, IL, J. Math. Soc. Japan 10 (1958), 94-105 and 235–248.Google Scholar
17. Murasugi, K., On alternating knots, Osaka Math. J. 12 (1960), 277–303.Google Scholar
18. Murasugi, K., On a certain subgroup of the group of an alternating link, Amer. J. Math. 85 (1963), 544–550.Google Scholar
19. Murasugi, K., On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387–422.Google Scholar
20. Murasugi, K., The commutator subgroups of the alternating knot groups, Proc. Amer. Math. Soc. 28 (1971), 237–241.Google Scholar
21. Neuwirth, L., Knot groups (Princeton IL Press, Princeton, N.J., 1965).Google Scholar
22. Schubert, H., Die eindeutige Zerlegbarkeit eines Knotens in Primknoten, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 3 (1949), 57–104.Google Scholar
23. Seifert, H., Uber das Geschlecht von Knoten, Math. Ann. 110 (1935), 571–592.Google Scholar
24. Stebe, P., Residual finiteness of a class of knot groups, Comm. Pure Appl. Math. 21 (1968), 563–583.Google Scholar