Ozsváth–Szabó and Rasmussen invariants of doubled knots

Abstract

Let $\nu$ be any integer-valued additive knot invariant that bounds the smooth 4–genus of a knot $K$, $\left|\nu \left(K\right)\right|\le g_4\left(K\right)$, and determines the 4–ball genus of positive torus knots, $\nu \left(T_{p,q}\right)=\left(p-1\right)\left(q-1\right)∕2$. Either of the knot concordance invariants of Ozsváth-Szabó or Rasmussen, suitably normalized, have these properties. Let $D_{\pm }\left(K,t\right)$ denote the positive or negative $t$–twisted double of $K$. We prove that if $\nu \left(D_{+}\left(K,t\right)\right)=\pm 1$, then $\nu \left(D_{-}\left(K,t\right)\right)=0$. It is also shown that $\nu \left(D_{+}\left(K,t\right)\right)=1$ for all $t\le \text{ TB}\left(K\right)$ and $\nu \left(D_{+}\left(K,t\right)\right)=0$ for all $t\ge -\text{ TB}\left(-K\right)$, where $\text{ TB}\left(K\right)$ denotes the Thurston-Bennequin number.

A realization result is also presented: for any $2g\times 2g$ Seifert matrix $A$ and integer $a$, $\left|a\right|\le g$, there is a knot with Seifert form $A$ and $\nu \left(K\right)=a$.