Foamho, an sl3-homology calculator

sl3-homology of the T(6,5)-torus knot:

q \ t012345678910111213141516171819
-78 (6,5)-torus knot
1
-761
-74
1
1
-722
1
1
1
1
-7012
1
2
1
-68121
1
3
2
-662
1
3
4
1
1
1
-6412
1
4
2
3
1
1
-6212
2
5
1
1
1
1
1
1
-6011
1
5
1
1
1
3
2
-58
1
43
2
3
1
1
-562
1
4
1
1
1
2
3
-541311
2
31
-5221
1
3
1
1
-5011
1
23 red: 2-torsion, green: 3-torsion, yellow: 5-torsion
-4811
1
21
-46
1
21
-4411
-4211
-401
-381

Computed in six minutes on an AMD Opteron, 2.2GHz, using 80MB of RAM.
Diagram drawn with the KnotTheory` package for Mathematica.

For details, see my paper sl3-foam homology calculations: arxiv.org/abs/1212.2553.

Download

foamho-1.1.tar.gz containing the source code and installation instructions in the README file. To compile, you will need at least the MPIR-library, which is used for rational and integral arbitrary precision arithmetic. Optionally, the PARI-library is used to obtain smith normal forms (not necessary if you only want to compute rational homology), and the PROCPS-library to display memory consumption (on Linux only).

foamho.exe, a windows executable compiled using minGW running on Windows XP. No additional libraries necessary, but it is not guaranteed to work on your system.

foamho, a linux executable. No additional libraries are necessary, but it is not guaranteed to work on your system.

Old version (1.0): foamho-1.0.tar.gz, foamho.exe, foamho.

Some results

Let us list all knots of twelve or less crossings for which the sl3- and sl3-concordance invariants differ (click on the knot to see his diagram on the Knot Atlas or on KnotInfo). In our normalisation, both invariants have a value of 2 for the right-handed trefoil knot.

Knot1012511n82 12n195 12n235 12n238 12n340 12n346 12n571 12n673 12n723 12n821
s222-2-4-202-2-22-2
s311-1-3-111-1-11-1
Last update: November 13th, 2017.