Heegaard Floer homology of large surgeries on knots

Here is the zipped directory with a Haskell program for calculating the hat version of Heegaard Floer homology of large surgeries on knots. The program is mostly the work of Damek Davis, with a few minor additions by Ciprian Manolescu.

The program takes as input a grid diagram for a knot K, and computes the generalized knot Floer homology H_*(A_s(K)) for s between 0 and m, where m is the maximal Alexander grading of the generators in the grid. Here A_s(K) is the complex C{max(i, j-s}=0} in the notation of [1]. Its homology is isomorphic to the hat Heegaard Floer homology of large surgery on K, in a Spin^c structure corresponding to s. (See [1], [2], [4].) It suffices to compute this homology for s between 0 and m, because for other values of s we can use the relations:

H_*(A_s(K)) = H_*(A_-s(K)) for all s

and

H_*(A_s(K)) = Z for s ≥ m.

The calculation of H_*(A_s(K)) is done using the model for knot Floer complexes coming from grid diagrams; see [5], [7], [8], [9]. In fact, the code is inspired from the program [6] for computing the usual knot Floer homology.

Instructions:

First, you should make sure you have the Haskell platform.
Next, you should have cabal installed and configured.
Now, you need to install two packages: Data.Vector and Data.Repa. Type cabal install Vector and cabal install Repa at your terminal. If you have trouble installing Repa, try typing cabal install repa-2.1.1.5 instead.
After that cd into the directory where ASHat.hs is located and type ghc --make -O2 ASHat.hs -XBangPatterns -XTypeOperators -XTypeSynonymInstances to compile the program.
Now try typing ./ASHat "[4, 0, 1, 2, 3]" "[1, 2, 3, 4, 0]". This is the left-handed trefoil. The first list corresponds to X's and the second corresponds to O's. The origin of the grid is the lower left hand corner.
The output is a list of the form:
[(0,[(0,2),(1,1)]),(1,[(0,1)])]
This means that for s=0 the homology is of rank 2 in relative grading 0 and rank 1 in relative grading 1. For s=1 (and in fact for any other nonzero s) the homology is of rank 1.

The program can deal with a grid diagram of size 8 in a few minutes. As such, it is not too useful -- the Heegaard Floer homology of large surgeries on small (e.g. alternating) knots can be computed by other methods [3]. However, it should be possible to extend the program to compute HF of large surgeries on links, first the hat version and (with more work) the plus and minus versions; there are several interesting and less-studied links of small arc index. With even more work, one can hope to extend the program to compute HF of any surgery on a link (hence of any 3-manifold), following [8], [9].

If you are interested in developing the program, contact us!

References:

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

[2] J. Rasmussen, Floer homology and knot complements, Ph.D. Thesis, Harvard University (2003).

[3] P. Ozsvath and Z. Szabo, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003), 225-254.

[4] P. Ozsvath and Z. Szabo, Knot Floer homology and integer surgeries, Algebr. Geom. Topol. 8 (2008), no. 1, 101-153.

[5] C. Manolescu, P. Ozsvath and S. Sarkar, A combinatorial description of knot Floer homology, Annals of Math. (2) 169 (2009), no. 2, 633-660.

[6] J. Baldwin and D. Gillam, Computations of Heegaard-Floer knot homology, preprint (2006).

[7] C. Manolescu, P. Ozsvath, Z. Szabo and D. Thurston, On combinatorial link Floer homology, Geom. Topol. 11 (2007), 2339-2412.

[8] C. Manolescu and P. Ozsvath, Heegaard Floer homology and integer surgeries on links, preprint (2010).

[9] C. Manolescu, P. Ozsvath and D. Thurston, Grid diagrams and Heegaard Floer invariants, preprint (2009).