Ozsvath-Szabo Tau-Invariant

The Ozsvath-Szabo Tau-Invariant was defined in Knot Floer homology and the four-ball genus by Peter Ozsvath and Zoltan Szabo which appeared in Geom. Topol. 7 (2003) 615-639. It satisfies the inequality |tau(K)| ≤ genus_4(K).

Heegaard-Floer homology knot homology groups were shown to be algorithmic in: Ciprian Manolescu, Peter Ozsvath, Sucharit Sarkar, "A combinatorial description of knot Floer homology," math.GT/0607691. This paper works with Z/2Z coefficients. In the paper "On combinatorial link Floer homology," (math.GT/0610559) Ciprian Manolescu, Peter Ozsvath, Zoltan Szabo, and Dylan Thurston resolve orientation issues and give an algorithm for computing Heegaard Floer knot and link invariants using Z coefficients. (They also give a combinatorial proof that these invariants are well defined.)

John Baldwin and W. D. Gillam have used this combinatorial approach to compute the Heegaard-Floer homology of many knots, including 11 crossing non-alternating knots. In particular, they prove that the value for 10_141 is 0. See: "Computations of Heegaard-Floer knot homology," math.GT/0610167

Further information on particular knots.