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Peripherally specified homomorphs of knot groups


Authors: Dennis Johnson and Charles Livingston
Journal: Trans. Amer. Math. Soc. 311 (1989), 135-146
MSC: Primary 57M25
DOI: https://doi.org/10.1090/S0002-9947-1989-0942427-5
MathSciNet review: 942427
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Abstract: Let $ G$ be a group and let $ \mu$ and $ \lambda$ be elements of $ G$. Necessary and sufficient conditions are presented for the solution of the following problem: Is there a knot $ K$ in $ {S^3}$ and a representation $ \rho :{\pi _1}({S^3} - K) \to G$ such that $ \rho (m) = \mu $ and $ \rho (l) = \lambda $, where $ m$ and $ l$ are the meridian and longitude of $ K$?


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

  • [1] K. Brown, Cohomology of groups, Springer-Verlag, New York, Heidelberg and Berlin, 1982. MR 672956 (83k:20002)
  • [2] P. Conner and E. Floyd, Differentiable periodic maps, Springer-Verlag, New York, 1964. MR 0176478 (31:750)
  • [3] A. Edmonds and C. Livingston, Symmetric representations of knot groups, Topology Appl. 18 (1984), 281-314. MR 769296 (86d:57003)
  • [4] F. Gonzalez-Acuna, Homomorphs of knot groups, Ann. of Math. (2) 102 (1975), 373-377. MR 0379671 (52:576)
  • [5] D. Johnson, Homomorphs of knot groups, Proc. Amer. Math. Soc. 78 (1980), 135-138. MR 548101 (80j:57004)
  • [6] -, Peripherally specified homomorphs of knot groups, preprint.
  • [7] R. Kirby, Problems in low-dimensional manifold theory, Proc. Sympos. Pure Math., vol. 32, Amer. Math. Soc., Providence, R. I., 1978, pp. 273-312. MR 520548 (80g:57002)
  • [8] P. Kutzko, On groups of finite weight, Proc. Amer. Math. Soc. 55 (1976), 279-280. MR 0399272 (53:3123)
  • [9] W. B. R. Lickorish, A representation of orientable combinatorial $ 3$-manifolds, Ann. of Math. 76 (1962), 531-538. MR 0151948 (27:1929)
  • [10] L. P. Neuwirth, Knot groups, Princeton Univ. Press, 1965. MR 0176462 (31:734)
  • [11] D. Rolfsen, Knots and links, Publish or Perish, Berkeley, Calif., 1976. MR 0515288 (58:24236)
  • [12] A. D. Wallace, Modifications and cobounding manifolds, Canad. J. Math 12 (1960), 503-528. MR 0125588 (23:A2887)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0942427-5
Article copyright: © Copyright 1989 American Mathematical Society

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