关于Livingston和Naik定义的$t_ν(K)$的不等式及其应用
Inequality on $t_ν(K)$ defined by Livingston and Naik and its applications
摘要：
令$D_+(K,t)$表示$K$的正$t$-双扭.对于一个固定的整数值加性和谐不变量$ν$，它限定了一个纽结的光滑四个亏格，并确定了正环面纽结的光滑四个亏格，Livingston和Naik定义$t_ν(K)$为最大整数$t$，使得$ν(D_+(K,t)) = 1$。让$K_1$和$K_2$是任何节点，然后我们证明了以下不等式：$t_ν(K_1) + t_ν(K_2) \leq t_ν(K_1 \# K_2) \leq min(t_ν(K_1) - t_ν(-K_2), t_ν(K_2) - t_ν(-K_1)).$作为一个应用程序，我们表明，$t_τ(K) \neq t_s(K)$为无穷多个节点，他们的差异可以是任意大，其中$t_τ(K)$（分别$t_s(K)$）是$t_ν(K)$时，$ν$是Ozváth-Szabó不变$τ$（分别当$ν$是规范化拉斯穆森$s$不变）。
英文摘要：
Let $D_+(K,t)$ denote the positive $t$-twisted double of $K$. For a fixed integer-valued additive concordance invariant $ν$ that bounds the smooth four genus of a knot and determines the smooth four genus of positive torus knots, Livingston and Naik defined $t_ν(K)$ to be the greatest integer $t$ such that $ν(D_+(K,t)) = 1$. Let $K_1$ and $K_2$ be any knots then we prove the following inequality : $t_ν(K_1) + t_ν(K_2) \leq t_ν(K_1 \# K_2) \leq min(t_ν(K_1) - t_ν(-K_2), t_ν(K_2) - t_ν(-K_1)).$ As an application we show that $t_τ(K) \neq t_s(K)$ for infinitely many knots and that their difference can be arbitrarily large, where $t_τ(K)$ (respectively $t_s(K)$) is $t_ν(K)$ when $ν$ is Ozváth-Szabó invariant $τ$ (respectively when $ν$ is normalized Rasmussen $s$ invariant).
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