Knot Floer Homology

Knot Floer homology is a filtered version of Heegaard Floer homology, due to Ozsváth and Szabó, and Rasmussen in references [1] and [2]. The `hat' version HFK-hat(S^3, Y) is a bigraded vector space over Z/2Z.

The data for HFK-hat displayed in this database, along with the invariants epsilon, nu, tau, and the L-space knot property were computed using a program written by Zoltan Szabó, available at [3]. The calculations were performed with Z/2Z coefficients.

Example to clarify notation: The Poincare polynomial for HFK-hat of the trefoil is

1a^(-1)m^(-2)+1a^(0)m^(-1)+1a^(1)m^(0)

This indicates three summands of rank one in bigradings (-1, -2), in (0, -1), and in (1, 0), where the convention is (Alexander, Maslov). Formatted as a vector, this data appears as

[1,-1,-2;1,0,-1;1,1,0]

Where the convention is [Rank, Alexander Maslov; ...]

References

[1] Ozsváth, P., Szabó, Z., "Holomorphic disks and knot invariants", Adv. Math. 186 (2004), no. 1, 58--116

[2] Rasmussen, J. "Floer homology and knot complements", Ph.D. thesis, Harvard University, 2003, arXiv:math.GT/0306378.

[3] Szabó, Z., Knot Floer Homology Calculator.