The Ozsvath-Szabo Tau-Invariant was defined in Knot Floer homology and the
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
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
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.