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

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.

concordance, cable, Rasmussen invariant, Ozsváth–Szabó concordance invariant
Mathematical Subject Classification 2000
Primary: 57M25
Received: 28 December 2009
Accepted: 5 January 2010
Published: 2 April 2010
Cornelia A Van Cott
Department of Mathematics
University of San Francisco
San Francisco, California