The Ozsvath-Szabo Tau-Invariant was defined in . It satisfies the inequality |tau(K)| ≤ genus4(K).
Heegaard-Floer homology groups were shown to be algorithmic in . This paper works with Z/2Z coefficients.
In , the authors 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.
Baldwin and Gillam have used this combinatorial approach to compute the Heegaard-Floer homology of many knots, including 11 crossing non-alternating knots . In particular, they show tau(10141) = 0.
 Baldwin, J. and Gillam, W. D., "Computations of Heegaard-Floer knot homology," math.GT/0610167.
 Manolescu, C., Ozsváth, P., and Sarkar, S., "A combinatorial description of knot Floer homology." Arxiv preprint.
 Manolescu, C., Ozsváth, P., Szabó, Z., and Thurston, D., "On combinatorial link Floer homology," math.GT/0610559.
 Ozsváth, P. and Szabó, Z., Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003), 615-639.