Rasmussen Invariant

Based on the generalization of Khovanov homology developed by Lee, Rasmussen [3] 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.

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

Many of the values of the s invariant listed in the table were adjusted to be correct with respect to orientation with the assistance of Dirk Schuetz and Se-Goo Kim.

References

[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.

[3] Rasmussen, J. A., "Khovanov homology and the slice genus," (2003), Arxiv preprint.