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Article overview
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Inequality on $t_
u(K)$ defined by Livingston and Naik and its applications | | JungHwan Park
; | | Date: |
15 Mar 2016 | | Abstract: | Let $D_+(K,t)$ denote the positive $t$-twisted double of $K$. For a fixed
integer-valued additive concordance invariant $
u$ that bounds the smooth four
genus of a knot and determines the smooth four genus of positive torus knots,
Livingston and Naik defined $t_
u(K)$ to be the greatest integer $t$ such that
$
u(D_+(K,t)) = 1$. Let $K_1$ and $K_2$ be any knots then we prove the
following inequality : $t_
u(K_1) + t_
u(K_2) leq t_
u(K_1 # K_2) leq
min(t_
u(K_1) - t_
u(-K_2), t_
u(K_2) - t_
u(-K_1)).$ As an application we
show that $t_ au(K)
eq t_s(K)$ for infinitely many knots and that their
difference can be arbitrarily large, where $t_ au(K)$ (respectively $t_s(K)$)
is $t_
u(K)$ when $
u$ is Ozv’{a}th-Szab’{o} invariant $ au$ (respectively
when $
u$ is normalized Rasmussen $s$ invariant). | | Source: | arXiv, 1603.4740 | | Services: | Forum | Review | PDF | Favorites | |
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