The Ozsváth-Szabó and Rasmussen Concordance Invariants are not Equal

Abstract

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 Ozsváth-Szabó Floer homology. The counterexamples are twisted Whitehead doubles of the (2, 2n + 1) torus knots.

A combinatorial description of knot Floer homology

“…This feature underscores its similarity with Rasmussen's concordance invariant s(K) [28] from Khovanov homology [12]. However, these two invariants are known to be linearly independent [11].…”

Knot Floer homology of Whitehead doubles

“…We should also remark that in the case where the companion knot is the .2; n/ torus knot, the above result follows from Hedden-Ording [12]. Indeed the main purpose of [12] was to show that .K/ does not equal half the Rasmussen concordance invariant, s.K/ [34].…”

“…In a related direction, the results of [19] and [12] indicate that there are two invariants associated to a knot:…”

“…Indeed the main purpose of [12] was to show that .K/ does not equal half the Rasmussen concordance invariant, s.K/ [34]. We believe that Whitehead doubles of knots with .K/ ¤ 0 will provide further examples of knots with 2 .K/ ¤ s.K/.…”

Turaev genus, knot signature, and the knot homology concordance invariants

“…Furthermore, if K is an alternating knot, then the Ozsváth-Szabó τ invariant [OS03b], the Rasmussen s invariant [Ras04], and the signature of K satisfy 2τ (K) = s(K) = −σ(K). Note that it took some effort to show that in general 2τ (K) and s(K) are not equal [HO08].…”