The Ozsváth-Szabó and Rasmussen Concordance Invariants are not Equal

By M. Hedden, Philip Ording. 2005

74 citations recorded by the source corpus.

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 (2, 2n + 1) torus knots.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/14ca301fee1e0a648def870ba63f36af202dc282

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Unoriented knot Floer homology and the unoriented four-ball genus (2015, background): “Observe that 2τ − s = 0 for all torus knots; any knot K with 2τ(K) 6= s(K) (the first examples of which were found by Hedden and Ording [5]) now completes the linear independence claim.”
Computations of Heegaard-Floer knot homology (2006, background): “On the other hand, Hedden and Ording showed in [13] that, for certain twisted Whitehead doubles of the right-handed trefoil, s ̸= 2τ .”
A combinatorial description of knot Floer homology (2006, background): “However, these two invariants are known to be linearly independent [9].”
A survey of Heegaard Floer homology (2013, background): “Now we state another conjecture, originally due to Ozsváth and Szabó, cf. Hedden and Ording [28].”
A remark on rasmussen's invariant of knots (2005, background): “The s-invariant was conjectured to be equal to twice the τ -invariant in Heegaard–Floer knot homology, however this is now known to be false in general [2].”
On knot Floer homology and cabling (2004, background): “dy of the Rasmussen concordance invariant, s(K). This latter invariant is deﬁned using Khovanov homology [35], and while it shares several important properties of τ the two invariants are independent [14]. Indeed, we expect the behavior of the corresponding functions sp(K,n) to be quite diﬀerent fro”
Man and machine thinking about the smooth 4-dimensional Poincaré conjecture (2009, background): “K/j, and when the s-invariant was discovered it was conjectured to coincide with , on the basis of small examples, although this was later proved false in [23].”
Knot Floer homology of Whitehead doubles (2006, methodology): “W hitehead doubling in the context ofOzsv ath-Szab o hom ology was rst studied by Eftekhary in [3].”