A program to compute HFhat and HFKhat


This is a beta version program I wrote to compute the hat version of the Heegaard Floer homology of a 3-manifold with Z2 coefficients, which realizes the algorithm in a paper of the author with Sucharit Sarkar. I will continue to update this program to have more features. Take your own risk to use this program.

The program is written in mathematica. Many part of the program should be compared with Ciprian's HF. The homology computation part is told to me by Peter Ozsvath.

Download: Here is the 2006.09.14 version. See here for the log of our update. Currently there are many checking codes since there are more than 1000 lines. When we believe there are no bugs, the performance will be improved with these codes removed.

Features to release soon: Start with a knot projection, compute the knot Floer homology for a knot in S^3.

What we do: We will compute the \widehat{HF} of an arbitrary three-manifold, or the \widehat{HFK} of a knot in an integer homology sphere. For a three-manifold other than a rational homology sphere, we will not give any grading information.

What we input: A heegaard diagram in a particular form. We require the Heegaard diagram has only disk regions AND is admissible. This is not hard to do, you just do finger moves to achieve this. If you input one with nondisk regions, the program will refuse to work! :) See here for the Heegaard diagram for the Poincare homology sphere, here (.nb mathematica file) for the usage of our package for this diagram, and here (.nb) for the computation for the trefoil knot. Click here for a list of Heegaard diagrams. And you could submit your Heegaard diagram to share with others.

What you get:
  
  • A nice Heegaard diagram;
      
  • \widehat{CF} and \widehat{HF} for the three manifold in each Spin^c structure;
      
  • \widehat{CFK} and \widehat{HFK} for the knot in each Spin^c structure;
      
  • The genus, fibredness, concordance tau invariant of the knot;
      
  • The corresponding contact invariant

    References
       1. Sucharit Sarkar and the author, A combinatorial description of some Heegaard Floer homologies, math.GT/0607777.
       2. Ciprian Manolescu, How to guess Heegaard Floer homology, http://www.columbia.edu/~cm2361/matlab.html.

    For comments and suggestions, please give me an email wang@math.berkeley.edu.

    Copyleft: J.J. 2006