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Groups and Complements of Knots

Published online by Cambridge University Press:  20 November 2018

C. D. Feustel  and
Wilbur Whitten
C. D. Feustel
Affiliation:
Virginia Polytechnic Institute and State University, Blacksburg, Virginia
Wilbur Whitten
Affiliation:
Virginia Polytechnic Institute and State University, Blacksburg, Virginia
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We investigate the extent to which knot groups determine knot manifolds and knot types. Let Ki(i = 1, 2) denote a tame knot in S3, let Ci denote a Ki-knot manifold, and assume that Π1(C1) ≈ Π1(C2). The first named author recently showed (in [6]) that, if C1 has no essential annulus, then C1 ≅ C2, and so K1 and K2 are equivalent, if K1 has property P.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 6 , 01 December 1978 , pp. 1284 - 1295
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Bing, R. H. and Martin, J. M., Cubes with knotted holes, Trans. Amer. Math. Soc. 155 (1971), 217–231.Google Scholar
2. Brown, E. M., Unknotting in M2 X I, Trans. Amer. Math. Soc. 123 (1966), 480–505.Google Scholar
3. Burde, G. and Zeischang, H., Eine Kennzeichung der Torusknoten, Math. Ann. 167 (1966), 169–176.Google Scholar
4. Cannon, J. W. and Feustel, C. D., Essential annuli and mobius bands in Mz, Trans. Amer. Math. Soc. 215 (1976), 219–239.Google Scholar
5. Feustel, C. D., A splitting theorem for closed orientable 3-manifolds, Topology 11 (1972), 151–158.Google Scholar
6. Feustel, C. D., On the torus theorem and its applications, Trans. Amer. Math. Soc. 217 (1976), 1–43.Google Scholar
7. Gonzalez-Acuna, F., Dehns construction on knots, Bol. Soc. Mat. Mexicana 15 (1970), 58–79.Google Scholar
8. Hilton, P. J. and Wylie, S., Homology theory, 4th ed. (Cambridge University Press, Cambridge, England, 1967).Google Scholar
9. Magnus, W., Karrass, A., and Solitar, D., Combinatorial group theory, Pure and Applied Mathematics 13 (New York, N.Y., (1966).Google Scholar
10. Neuwirth, L. P., Knot groups, Annals of Mathematics Studies 56 (Princeton University Press, Princeton, N.J., (1965).Google Scholar
11. Noga, D., Uber den Aussenraum von Produktknoten una die Bedeutung der Fixgruppen, Math. Z. 101 (1967), 131–141.Google Scholar
12. Reidemeister, K., Knotentheorie (Chelsea Publishing Company, New York, N.Y., 1948).Google Scholar
13. Schubert, H., Die eindeutige Zerlegbarkeit eines Knotens in Primknoten, S.-B. Heidelberger Akad. Wiss. Math.-Natur. Kl. 3 (1949), 57–105.Google Scholar
14. Schubert, H., Knoten und Vollringe, Acta Math. 00 (1953), 131–286.Google Scholar
15. Scott, G. P., On sufficiently large 3-manifolds, Quart. J. Math. (Oxford Ser.) 23 (1972), 159–172.Google Scholar
16. Simon, J., Some classes of knots with property P, Topology of Manifolds, Proc. the Univ. of Georgia Inst., 1969 (Markham Publishing Company, Chicago, 1970), pp. 195–199.Google Scholar
17. Simon, J., An algebraic classification of knots in S3, Ann. of Math. 07 (1973), 1-13, 18. On the problems of determining knots by their complements and knot complements by their groups, Proc. Amer. Math. Soc. 57 (1976), 140–142.Google Scholar
19. Stallings, J., Group theory and three-dimensional manifolds, Yale Mathematical Monographs 4 (Yale University Press, New Haven, Conn., (1971).Google Scholar
20. Waldhausen, F., Gruppen mit Zentrum und 3-dimensionale Mannigfaltigkeiten, Topology G (1967), 505–517.Google Scholar
21. Waldhausen, F., On irreducible 3-manifolds which are sufficiently large, Ann. of Math. 87 (1968), 56–88.Google Scholar
22. Waldhausen, F., On the determination of some bounded 3-manifolds by their fundamental groups alone, Proc. Internat. Sympos. on Topology and its Applications, (Herieg-Novi, Yugoslavia, Aug. 25-31, (1968), Belgrad, 1969), 331–332.Google Scholar
23. Waldhausen, F., Recent results on sufficiently large 3-manifolds, Proc. AMS Summer School on Alg. and Geo. Topology, (Stanford University, Stanford, California, Aug., 1976), to appear.Google Scholar
24. Whitehead, J. H. C., Combinatorial homotopy, I, Bull. Amer. Math. Soc. 55 (1959), 213–245.Google Scholar
25. Whitten, W., Algebraic and geometric characterizations of knots, Invent. Math. 26 (1974). 259–270.Google Scholar