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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?)

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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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