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Ozsváth-Szabó And Rasmussen Invariants Of Cable Knots, Cornelia A. Van Cott

Mathematics

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 …