Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Algebraic meridians of knot groups


Author: Chichen M. Tsau
Journal: Trans. Amer. Math. Soc. 294 (1986), 733-747
MSC: Primary 57M25; Secondary 57M05
DOI: https://doi.org/10.1090/S0002-9947-1986-0825733-1
MathSciNet review: 825733
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We propose the conjecture that every automorphism of a knot group preserves the meridian up to inverse and conjugation. We establish the conjecture for all composite knots, all torus knots, most cable knots, and at most one exception for hyperbolic knots; moreover we prove that the Property P Conjecture implies our conjecture. We also investigate hyperbolic knots in more detail, and give an example of figure-eight knot group and its automorphisms.


References [Enhancements On Off] (What's this?)

  • [1] E. M. Brown, Unknotting in $ {M^2} \times I$, Trans. Amer. Math. Soc. 123 (1966), 480-505. MR 0198482 (33:6640)
  • [2] C. Feustel and W. Whitten, Groups and complements of knots, Canad. J. Math. 30 (1978), 1284-1295. MR 511562 (80b:57004)
  • [3] K. Johannson, Homotopy equivalences of $ 3$-manifolds with boundaries, Lecture Notes in Math., Vol. 761, Springer-Verlag, Berlin and New York, 1979. MR 551744 (82c:57005)
  • [4] R. Litherland, Surgery on knots in solid tori. II, J. London Math. Soc. 22 (1980), 559-569. MR 596334 (82d:57002)
  • [5] W. Magnus, Untersuchungen über einige unendliche diskontinuierliche Gruppen, Math. Ann. 105 (1931), 52-74. MR 1512704
  • [6] A. Marsden, The geometry of finitely generated Kleinian groups, Ann. of Math. (2) 99 (1974), 383-462. MR 0349992 (50:2485)
  • [7] R. Riley, A quadratic parabolic group, Math. Proc. Cambridge Philos. Soc. 77 (1975), 281-288. MR 0412416 (54:542)
  • [8] -, Automorphisms of excellent link groups, preprint.
  • [9] H. Schubert, Knoten und Vollringe, Acta Math. 90 (1953), 131-286. MR 0072482 (17:291d)
  • [10] O. Schreier, Über die Gruppen $ {A^a}{B^b} = 1$, Abh. Math. Sem. Univ. Hamburg 3 (1923), 167-169.
  • [11] J. Simon, Some classes of knots with Property $ {\text{P}}$, Topology of Manifolds, Markham, Chicago, Ill., 1970. MR 0278288 (43:4018b)
  • [12] W. Thurston, The geometry and topology of $ 3$-manifolds, Lecture notes, Princeton Univ.
  • [13] F. Waldhausen, On irreducible $ 3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56-88. MR 0224099 (36:7146)
  • [14] -, Gruppen mit Zentrum und $ 3$-dimensionale Mannigfaltigkeiten, Topology 6 (1967), 505-517. MR 0236930 (38:5223)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57M25, 57M05

Retrieve articles in all journals with MSC: 57M25, 57M05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0825733-1
Keywords: Knot manifold, knot group, presentation, automorphism of knot group, torus knot, composite knot, cable knot, hyperbolic knot, surgery manifold
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society