## Algebraic & Geometric Topology

### Ozsváth–Szabó and Rasmussen invariants of doubled knots

#### Abstract

Let $ν$ be any integer-valued additive knot invariant that bounds the smooth 4–genus of a knot $K$, $|ν(K)|≤g4(K)$, and determines the 4–ball genus of positive torus knots, $ν(Tp,q)=(p−1)(q−1)∕2$. Either of the knot concordance invariants of Ozsváth-Szabó or Rasmussen, suitably normalized, have these properties. Let $D±(K,t)$ denote the positive or negative $t$–twisted double of $K$. We prove that if $ν(D+(K,t))=±1$, then $ν(D−(K,t))=0$. It is also shown that $ν(D+(K,t))=1$ for all and $ν(D+(K,t))=0$ for all , where denotes the Thurston-Bennequin number.

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

#### Article information

Source
Algebr. Geom. Topol., Volume 6, Number 2 (2006), 651-657.

Dates
Received: 26 February 2006
Accepted: 2 March 2006
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796539

Digital Object Identifier
doi:10.2140/agt.2006.6.651

Mathematical Reviews number (MathSciNet)
MR2240910

Zentralblatt MATH identifier
1096.57010

#### Citation

Livingston, Charles; Naik, Swatee. Ozsváth–Szabó and Rasmussen invariants of doubled knots. Algebr. Geom. Topol. 6 (2006), no. 2, 651--657. doi:10.2140/agt.2006.6.651. https://projecteuclid.org/euclid.agt/1513796539