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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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Concordance invariants of doubled knots and blowing up
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by Se-Goo Kim and Kwan Yong Lee PDF
Proc. Amer. Math. Soc. 147 (2019), 1781-1788 Request permission

Abstract:

Let $\nu$ be either the Ozsváth–Szabó $\tau$–invariant or the Rasmussen $s$–invariant, suitably normalized. For a knot $K$, Livingston and Naik defined the invariant $t_\nu (K)$ to be the minimum of $k$ for which $\nu$ of the $k$–twisted positive Whitehead double of $K$ vanishes. They proved that $t_\nu (K)$ is bounded above by $-TB(-K)$, where $TB$ is the maximal Thurston–Bennequin number. We use a blowing-up process to find a crossing change formula and a new upper bound for $t_\nu$ in terms of the unknotting number. As an application, we present infinitely many knots $K$ such that the difference between Livingston–Naik’s upper bound $-TB(-K)$ and $t_\nu (K)$ can be arbitrarily large.
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Additional Information
  • Se-Goo Kim
  • Affiliation: Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, Seoul, 02447 Republic of Korea
  • MR Author ID: 610250
  • ORCID: 0000-0002-8874-9408
  • Email: sgkim@khu.ac.kr
  • Kwan Yong Lee
  • Affiliation: Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, Seoul, 02447 Republic of Korea
  • Email: lky_0705@naver.com
  • Received by editor(s): November 29, 2017
  • Received by editor(s) in revised form: June 1, 2018, and July 6, 2018
  • Published electronically: January 9, 2019
  • Additional Notes: This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2015R1D1A1A01058384).
  • Communicated by: David Futer
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 1781-1788
  • MSC (2010): Primary 57M25; Secondary 57N70
  • DOI: https://doi.org/10.1090/proc/14320
  • MathSciNet review: 3910442