Ozsvath-Szabo Tau-Invariant

The Ozsvath-Szabo Tau-Invariant was defined in [4]. It satisfies the inequality |tau(K)| ≤ genus4(K).

Heegaard-Floer homology groups were shown to be algorithmic in [2]. This paper works with Z/2Z coefficients.

In [3], 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.

Specific Knots

Baldwin and Gillam have used this combinatorial approach to compute the Heegaard-Floer homology of many knots, including 11 crossing non-alternating knots [1]. In particular, they show tau(10141) = 0.

References

[1] Baldwin, J. and Gillam, W. D., "Computations of Heegaard-Floer knot homology," math.GT/0610167.

[2] Manolescu, C., Ozsváth, P., and Sarkar, S., "A combinatorial description of knot Floer homology." Arxiv preprint.

[3] Manolescu, C., Ozsváth, P., Szabó, Z., and Thurston, D., "On combinatorial link Floer homology," math.GT/0610559.

[4] Ozsváth, P. and Szabó, Z., Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003), 615-639.