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,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.

Keywords

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

Mathematical Subject Classification

Primary: 57M25

References

Publication

Received: 28 December 2009
Accepted: 5 January 2010
Published: 2 April 2010

Authors