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Longitude surgery on genus $ 1$ knots


Author: Howard Lambert
Journal: Proc. Amer. Math. Soc. 63 (1977), 359-362
MSC: Primary 55A25
DOI: https://doi.org/10.1090/S0002-9939-1977-0438322-8
MathSciNet review: 0438322
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Abstract: Let $ l(K)$ be the closed 3-manifold obtained by longitude surgery on the knot manifold K. Let C be the cube with holes obtained by removing an open regular neighborhood of a minimal spanning surface in K. The main result of this paper is that if K is of genus 1 and the longitude of K is in each term of the lower central series for $ {\Pi _1}(C)$, then $ l(K)$ is not homeomorphic to the connected sum of $ {S^1} \times {S^2}$ and a homotopy 3-sphere. In particular, this implies we cannot obtain the connected sum of $ {S^1} \times {S^2}$ and a homotopy 3-sphere by longitude surgery on any pretzel knot of genus 1.


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

  • [1] P. S. Alexandrov, Combinatorial topology, Vol. 3, Graylock Press, Albany, N. Y., 1960. MR 22 #4056. MR 0113218 (22:4056)
  • [2] R. H. Fox, A quick trip through knot theory, Topology of 3-Manifolds and Related Topics (Proc. Univ. Georgia Inst., 1961), Prentice-Hall, Englewood Cliffs, N. J., 1962, pp. 120-167. MR 25 #3522. MR 0140099 (25:3522)
  • [3] W. Haken, Some results on surfaces in 3-manifolds, Studies in Modern Topology, Math. Assoc. Amer.; distributed by Prentice-Hall, Englewood Cliffs, N. J., 1968, pp. 39-98. MR 36 #7118. MR 0224071 (36:7118)
  • [4] William Jaco and D. R. McMillan, Jr., Retracting three-manifolds onto finite graphs, Illinois J. Math. 14 (1970), 150-158. MR 41 #1026. MR 0256370 (41:1026)
  • [5] W. Magnus, A. Karrass, and D. Solitar, Combinatorial group theory, Interscience, New York, 1966. MR 34 #7617.
  • [6] L. Moser, On the impossibility of obtaining $ {S^2} \times {S^1}$ by elementary surgery along a knot, Pacific J. Math. 53 (1974), 519-523. MR 50 #11215. MR 0358756 (50:11215)
  • [7] K. Reidemeister, Knotentheorie, Chelsea, New York, 1948.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0438322-8
Keywords: Longitude surgery, genus 1 knots, pretzel knots, connected sum of $ {S^1} \times {S^2}$ and a homotopy 3-sphere
Article copyright: © Copyright 1977 American Mathematical Society

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