#### Volume 6, issue 2 (2006)

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Ozsváth–Szabó and Rasmussen invariants of doubled knots

### Charles Livingston and Swatee Naik

Algebraic & Geometric Topology 6 (2006) 651–657
 arXiv: math.GT/0505361
##### Abstract

Let $\nu$ be any integer-valued additive knot invariant that bounds the smooth 4–genus of a knot $K$, $|\nu \left(K\right)|\le {g}_{4}\left(K\right)$, 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 ${D}_{±}\left(K,t\right)$ denote the positive or negative $t$–twisted double of $K$. We prove that if $\nu \left({D}_{+}\left(K,t\right)\right)=±1$, 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 and $\nu \left({D}_{+}\left(K,t\right)\right)=0$ for all , where denotes the Thurston-Bennequin number.

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

##### Keywords
doubled knot, Ozsvath-Szabo invariant, Rasmussen invariant
Primary: 57M27
Secondary: 57M25