Ozsváth–Szabó and Rasmussen invariants of cable knots

Van Cott, Cornelia A

doi:10.2140/agt.2010.10.825

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Ozsváth–Szabó and Rasmussen invariants of cable knots

Cornelia A Van Cott

Algebraic & Geometric Topology 10 (2010) 825–836

DOI: 10.2140/agt.2010.10.825

Abstract

We study the behavior of the Ozsváth–Szabó and Rasmussen knot concordance invariants $τ$ and $s$ on $K_{m, n}$ , 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 $K_{m, n}$ 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 $τ$ on $(m, m r + 1)$ –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.

Keywords

concordance, cable, Rasmussen invariant, Ozsváth–Szabó concordance invariant

Mathematical Subject Classification 2000

Primary: 57M25

References

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

Volume 10, issue 2 (2010)