Knot Floer complexes:
I have written Python code to compute the immersed curves representing the UV=0 knot Floer complex for knots in S^3.
The code is
available here .
The process relies on
Zoltan Szabo's program for computing the UV=0 knot Floer complex from a PD code for a knot, slighly modified to output the differential.
My code reads the output of that program and implements the arrow sliding algorithm described in my papers to reduce the complex to a normal form and represent it by an immersed multicurve.
Szabo's program has since been incorporated into
SnapPy; in the future I hope to update my Python module to interface directly with SnapPy.
For convenience I have computed the knot Floer complexes for all prime knots through 16 crossings and saved the results in the following .txt files. I hope this may serve as a useful and convenient source of examples for others.
Knot Floer complexes by crossing number:
3 ,
4 ,
5 ,
6 ,
7 ,
8 ,
9 ,
10 ,
11a ,
11n ,
12a ,
12n ,
13a ,
13n ,
14a ,
14n ,
15a ,
15n
16n-0 ,
16n-1 ,
16n-2 ,
16n-3 ,
16n-4 ,
16n-5 ,
16n-6 ,
16n-7 ,
16n-8 ,
16n-9 ,
16n-10
Each line of each file stores the knot Floer complex of a single knot, so for instance the 133rd line of 11n.txt represents the knot Floer complex for the knot 11n_133. The 16n knots are split into groups of 100,000, so the kth line of the file 16n-x corresponds to the knot 16n_(100000*x + k).
As an example, here is the data for 11n_133:
[2, 1, -1, 0, 1, -1, -2], (3,2), (2,2)^2, (-2,2)^2, (-3,2)
This line of text encodes a graded immersed multicurve in the infinite punctured strip S = [-1/2,1/2]xR with punctures at points (0,n+1/2) as follows:
- The first list of integers represents the distinguished curve component gamma_0, which is actually an immersed arc starting on the left side of the strip and ending on the right end of the strip.
The integers record the height each time the curve crosses the vertical line {0}xR through the punctures.
Each intersection with this vertical line corresponds to a generator of CFK, and the height gives the Alexander grading.
The Maslov grading can also be extracted, up to an overall shift, from each component of the curve.
The shift for the distinguished component is set by the fact that the first generator has Maslov grading 0.
- All other components of the immersed curve are closed curves.
In general these are also represented by lists of integers recording the heights of successive crossings of the curve with the vertical line through the punctures,
though now the starting point is arbitrary so these lists are only well-defined up to cyclic permutation. The curve is assumed to be moving rightward at the intersection corresponding to the first entry in the list.
Each such list is accompanied by an integer specifying the Maslov grading of the generator corresponding to the first integer in the list.
-
By far the most common closed components are simple figure eight components, which have the form
([a, a+1, a, a-1], m)
These components have four generators, with two in some Alexander grading and one in the gradings above and below this grading.
The delta grading (i.e. Alexander grading - Maslov grading) is the same for all four generators.
For these components we use a shorthand, representing them by a tuple (a, d), where a is the middle Alexander grading and d is the delta grading.
This shorthand is convenient because simple figure eight components are exceedingly common; in fact there is only one knot through 15 crossings whose immersed curve contains a closed component
that is not a simple figure eight component (the knot is 15n_166130).
-
Exponents indicate multiple copies of the same curve.
So the immersed curve associated with the knot 11n_133 has, in addition to gamma_0, six simple figure eight components.
There is one simple figure eight component centered at height 3, two centered at height 2, two centered at height -2, and one centered at height -3.
All of the figure eight components have delta grading 2.
Cosmetic surgeries:
As an application of the computations above and of obstructions I found to purely cosmetic surgeries,
I verified the cosmetic surgery conjecture for all knots with 16 our fewer crossings. The code used for this computation
can also be found in the
GitHub repository for the immersed curves code.
HF-hat of graph manifolds:
My PhD thesis included using bordered Floer homology to give an algorithm for computing Heegaard Floer homology of graph manifolds.
I wrote Python module implementing this; it computes HF-hat for an
arbitrary graph manifold given a weighted graph representing the manifold:
HFhat_graph_manifolds