#
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).