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Concordance invariants of doubled knots and blowing up


Authors: Se-Goo Kim and Kwan Yong Lee
Journal: Proc. Amer. Math. Soc. 147 (2019), 1781-1788
MSC (2010): Primary 57M25; Secondary 57N70
DOI: https://doi.org/10.1090/proc/14320
Published electronically: January 9, 2019
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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
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

DOI: https://doi.org/10.1090/proc/14320
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
Article copyright: © Copyright 2019 American Mathematical Society