Rasmussen Invariant

Based on the generalization of Khovanov homology developed by Lee, Rasmussen defined the so-called "s" invariant and proved that it defines a homomorphism from the concordance group to Z. Furthermore, it provides a lower bound for the smooth 4-genus of a knot. For alternating knots, it equals the signature.

Another "s" invariant related to Khovanov homology has been defined by Bar Natan, but its definition depends on a conjecture regarding Khovanov homology that has yet to be verified.

Hedden and Ording constructed examples to show that the Rasmussen and Ozsvath-Szabo invariants are distinct [2], and Livingston used those to construct Alexander polynomial one knots with distinct Rasmussen and Ozsvath-Szabo invariants [1].


[1] Livingston, C., "Slice knots with distinct Ozsvath-Szabo and Rasmussen Invariants," Arxiv preprint.

[2] Hedden, M. and Ording, P., "The Ozsvath-Szabo and Rasmussen concordance invariants are not equal," Arxiv preprint.