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Inequality on $ t_\nu(K)$ defined by Livingston and Naik and its applications


Author: JungHwan Park
Journal: Proc. Amer. Math. Soc. 145 (2017), 889-891
MSC (2010): Primary 57M25
DOI: https://doi.org/10.1090/proc/13306
Published electronically: August 17, 2016
MathSciNet review: 3577888
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Abstract: Let $ D_+(K,t)$ denote the positive $ t$-twisted double of $ K$. For a fixed integer-valued additive concordance invariant $ \nu $ that bounds the smooth four genus of a knot and determines the smooth four genus of positive torus knots, Livingston and Naik defined $ t_\nu (K)$ to be the greatest integer $ t$ such that $ \nu (D_+(K,t)) = 1$. Let $ K_1$ and $ K_2$ be any knots; then we prove the following inequality: $ t_\nu (K_1) + t_\nu (K_2) \leq t_\nu (K_1 \char93 K_2) \leq min(t_\nu (K_1) - t_\nu (-K_2), t_\nu (K_2) - t_\nu (-K_1)).$ As an application we show that $ t_\tau (K) \neq t_s(K)$ for infinitely many knots and that their difference can be arbitrarily large, where $ t_\tau (K)$ (respectively $ t_s(K)$) is $ t_\nu (K)$ when $ \nu $ is an Ozváth-Szabó invariant $ \tau $ (respectively when $ \nu $ is a normalized Rasmussen $ s$ invariant).


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Additional Information

JungHwan Park
Affiliation: Department of Mathematics, Rice University MS-136, 6100 Main Street, P.O. Box 1892, Houston, Texas 77251-1892
Email: jp35@rice.edu

DOI: https://doi.org/10.1090/proc/13306
Received by editor(s): April 12, 2016
Published electronically: August 17, 2016
Additional Notes: The author was partially supported by National Science Foundation grant DMS-1309081
Communicated by: David Futer
Article copyright: © Copyright 2016 American Mathematical Society