#### Volume 10, issue 2 (2010)

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Ozsváth–Szabó and Rasmussen invariants of cable knots

### Cornelia A Van Cott

Algebraic & Geometric Topology 10 (2010) 825–836
##### Abstract

We study the behavior of the Ozsváth–Szabó and Rasmussen knot concordance invariants $\tau$ and $s$ on ${K}_{m,n}$, the $\left(m,n\right)$–cable of a knot $K$ where $m$ and $n$ are relatively prime. We show that for every knot $K$ and for any fixed positive integer $m$, both of the invariants evaluated on ${K}_{m,n}$ differ from their value on the torus knot ${T}_{m,n}$ by fixed constants for all but finitely many $n>0$. Combining this result together with Hedden’s extensive work on the behavior of $\tau$ on $\left(m,mr+1\right)$–cables yields bounds on the value of $\tau$ on any $\left(m,n\right)$–cable of $K$. In addition, several of Hedden’s obstructions for cables bounding complex curves are extended.

##### Keywords
concordance, cable, Rasmussen invariant, Ozsváth–Szabó concordance invariant
Primary: 57M25
##### Publication
Received: 28 December 2009
Accepted: 5 January 2010
Published: 2 April 2010
##### Authors
 Cornelia A Van Cott Department of Mathematics University of San Francisco San Francisco, California 94117