Ozsvath-Szabo and Rasmussen invariants of cable knots

Abstract

We study the behavior of the Ozsvath-Szabo and Rasmussen knot concordance invariants tau 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 a fixed constant for all but finitely many n>0. Combining this result together with Hedden's extensive work on the behavior of tau on (m,mr+1)-cables yields bounds on the value of tau on any (m,n)-cable of K. In addition, several of Hedden's obstructions for cables bounding complex curves are extended.

Several corollaries are added, and the exposition is extended

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• Floer homology and knot complements
• Knot Floer homology and the four-ball genus
• Heegaard Floer homology and alternating knots
• Algebraic functions and closed braids
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• Bounds for the Thurston-Bennequin number from Floer homology
• On knot Floer homology and cabling
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• On closed 3-braids with unknotting number one
• On the Upsilon invariant of cable knots
• Some bounds for the knot Floer $\tau$-invariant of satellite knots
• The Rasmussen invariant of a homogeneous knot
• Rasmussen invariants of Whitehead doubles and other satellites
• Simplified Khovanov-Rozansky Chain Complexes of Open 2-braids
• A cabling formula for ${\nu }^{+}$ invariant
• Turaev genus, knot signature, and the knot homology concordance invariants
• A Combinatorial Description of the Knot Concordance Invariant Epsilon