Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Longitude surgery on genus $ 1$ knots

Author: Howard Lambert
Journal: Proc. Amer. Math. Soc. 63 (1977), 359-362
MSC: Primary 55A25
MathSciNet review: 0438322
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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. Aleksandrov, Combinatorial topology, Graylock Press, Albany, N.Y., 1960. MR 0113218
  • [2] R. H. Fox, A quick trip through knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 120–167. MR 0140099
  • [3] Wolfgang 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 0224071
  • [4] William Jaco and D. R. McMillan Jr., Retracting three-manifolds onto finite graphs, Illinois J. Math. 14 (1970), 150–158 (German). MR 0256370
  • [5] W. Magnus, A. Karrass, and D. Solitar, Combinatorial group theory, Interscience, New York, 1966. MR 34 #7617.
  • [6] Louise E. Moser, On the impossibility of obtaining 𝑆²×𝑆¹ by elementary surgery along a knot, Pacific J. Math. 53 (1974), 519–523. MR 0358756
  • [7] K. Reidemeister, Knotentheorie, Chelsea, New York, 1948.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55A25

Retrieve articles in all journals with MSC: 55A25

Additional Information

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

American Mathematical Society