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, and Livingston used those to construct Alexander polynomial one knots with distinct Rasmussen and Ozsvath-Szabo invariants.


References

M. Hedden and P. Ording, The Ozsvath-Szabo and Rasmussen concordance invariants are not equal, arxiv.org/math.GT/0512348

C. Livingston, Slice knots with distinct Ozsvath-Szabo and Rasmussen Invariants, arxiv.org/math.GT/0602631 Authors: