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Transactions of the American Mathematical Society

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



Surgery on a curve in a solid torus

Author: J. P. Neuzil
Journal: Trans. Amer. Math. Soc. 204 (1975), 385-406
MSC: Primary 55A25; Secondary 57A10
MathSciNet review: 0367970
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Abstract: We consider the following surgery question: If a regular neighborhood of a polyhedral knot in a solid torus is removed and then sewn back differently, what manifold results? We consider two classes of knots, torus knots and what we call doubly twisted knots. We obtain some related results on surgery on knots in $ {S^3}$.

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  • [1] R. H. Bing, Some aspects of the topology of $ 3$-manifolds related to the Poincaré conjecture, Lectures on Modern Math., vol. II, Wiley, New York, 1964, pp. 93-128. MR 30 #2474. MR 0172254 (30:2474)
  • [2] R. H. Bing and J. M. Martin, Cubes with knotted holes, Trans. Amer. Math. Soc. 155 (1971), 217-231. MR 43 #4018a. MR 0278287 (43:4018a)
  • [3] H. S. M. Coxeter, The groups determined by the relations $ {S^l} = {T^m} = {({S^{ - 1}}{T^{ - 1}}ST)^P} = 1$, Duke Math. J. 2 (1936), 61-73. MR 1545905
  • [4] -, The abstract groups $ {G^{m,n,p}}$, Trans. Amer. Math. Soc. 45 (1939), 73-150. MR 1501984
  • [5] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Springer-Verlag, New York, 1972. MR 0349820 (50:2313)
  • [6] J. Hempel, A simply connected $ 3$-manifold is $ {S^3}$ if it is the sum of a solid torus and the complement of the torus knot, Proc. Amer. Math. Soc. 15 (1964), 154-158. MR 28 #599. MR 0157365 (28:599)
  • [7] Jonathan Simon, Methods for proving that certain classes of knots have property $ P$, Ph. D. Thesis, University of Wisconsin, Madison, Wis., 1969.
  • [8] -, Some classes of knots with property $ P$, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, Ill., 1970, pp. 195-199. MR 43 #4018b.

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Article copyright: © Copyright 1975 American Mathematical Society

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