Ozsvath-Szabo and Rasmussen invariants of cable knots

TL;DR

This paper investigates how Ozsvath-Szabo and Rasmussen invariants behave on cable knots, revealing their near-constant difference from torus knots and providing bounds and obstructions related to complex curves.

Contribution

It establishes the near-constant difference of invariants on cable knots compared to torus knots and extends obstructions for cables bounding complex curves.

Findings

Invariants differ from torus knots by a fixed constant for large n.

Bounds on tau for (m,n)-cables of any knot K.

Extended obstructions for cables bounding complex curves.

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.