The Ozsv{\'a}th-Szab{\'o} and Rasmussen concordance invariants are not equal

TL;DR

This paper provides counterexamples to a conjecture in knot theory, showing that two important concordance invariants derived from different homology theories are not always equal, thus challenging previous assumptions.

Contribution

The paper constructs explicit counterexamples using twisted Whitehead doubles of torus knots, disproving Rasmussen's conjecture about the equality of the invariants.

Findings

Counterexamples to Rasmussen's conjecture are presented.

The invariants from Khovanov and Ozsváth-Szabó Floer homology are not always equal.

Twisted Whitehead doubles of (2,2n+1) torus knots serve as the counterexamples.

Abstract

In this paper we present several counterexamples to Rasmussen's conjecture that the concordance invariant coming from Khovanov homology is equal to twice the invariant coming from Ozsv{\'a}th-Szab{\'o} Floer homology. The counterexamples are twisted Whitehead doubles of the (2,2n+1) torus knots.