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

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 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear Contacts ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
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