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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Knot concordance and homology cobordism
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by Tim D. Cochran, Bridget D. Franklin, Matthew Hedden and Peter D. Horn PDF
Proc. Amer. Math. Soc. 141 (2013), 2193-2208 Request permission

Abstract:

We consider the question: “If the zero-framed surgeries on two oriented knots in $S^3$ are $\mathbb {Z}$-homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?” We show that this question has a negative answer in the smooth category, even for topologically slice knots. To show this we first prove that the zero-framed surgery on $K$ is $\mathbb {Z}$-homology cobordant to the zero-framed surgery on many of its winding number one satellites $P(K)$. Then we prove that in many cases the $\tau$ and $s$-invariants of $K$ and $P(K)$ differ. Consequently neither $\tau$ nor $s$ is an invariant of the smooth homology cobordism class of the zero-framed surgery. We also show that a natural rational version of this question has a negative answer in both the topological and smooth categories by proving similar results for $K$ and its $(p,1)$-cables.
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Additional Information
  • Tim D. Cochran
  • Affiliation: Department of Mathematics, Rice University, Houston, Texas 77251
  • Email: cochran@math.rice.edu
  • Bridget D. Franklin
  • Affiliation: Department of Mathematics, Rice University, Houston, Texas 77251
  • Email: bridget.franklin@alumni.rice.edu
  • Matthew Hedden
  • Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
  • MR Author ID: 769768
  • Email: mhedden@math.msu.edu
  • Peter D. Horn
  • Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
  • Address at time of publication: Department of Mathematics, Syracuse University, Syracuse, New York 13244
  • MR Author ID: 855878
  • Email: pdhorn@math.columbia.edu, pdhorn@syr.edu
  • Received by editor(s): November 22, 2010
  • Received by editor(s) in revised form: April 13, 2011, September 28, 2011, and September 30, 2011
  • Published electronically: January 11, 2013
  • Additional Notes: The first author was partially supported by National Science Foundation DMS-1006908
    The second author was partially supported by Nettie S. Autry Fellowship
    The third author was partially supported by NSF DMS-0906258
    The fourth author was partially supported by NSF Postdoctoral Fellowship DMS-0902786
  • Communicated by: Daniel Ruberman
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 2193-2208
  • MSC (2010): Primary 57N70, 57M25
  • DOI: https://doi.org/10.1090/S0002-9939-2013-11471-1
  • MathSciNet review: 3034445