## Algebraic & Geometric Topology

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

Cornelia A Van Cott

#### Abstract

We study the behavior of the Ozsváth–Szabó and Rasmussen knot concordance invariants $τ$ and $s$ on $Km,n$, the $(m,n)$–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 $Km,n$ differ from their value on the torus knot $Tm,n$ by fixed constants for all but finitely many $n>0$. Combining this result together with Hedden’s extensive work on the behavior of $τ$ on $(m,mr+1)$–cables yields bounds on the value of $τ$ on any $(m,n)$–cable of $K$. In addition, several of Hedden’s obstructions for cables bounding complex curves are extended.

#### Article information

Source
Algebr. Geom. Topol., Volume 10, Number 2 (2010), 825-836.

Dates
Accepted: 5 January 2010
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715115

Digital Object Identifier
doi:10.2140/agt.2010.10.825

Mathematical Reviews number (MathSciNet)
MR2629765

Zentralblatt MATH identifier
1195.57035

#### Citation

Van Cott, Cornelia A. Ozsváth–Szabó and Rasmussen invariants of cable knots. Algebr. Geom. Topol. 10 (2010), no. 2, 825--836. doi:10.2140/agt.2010.10.825. https://projecteuclid.org/euclid.agt/1513715115

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