knotkit is a C++ software package written by Cotton Seed (cseed@math.princeton.edu) for computing some knot and manifold invariants appearing in low-dimensional topology. Other contributors include Josh Batson.
TABLE OF CONTENTS
INTRO
BUILDING
USAGE
UPCOMING CHANGES
FOR DEVELOPERS
INTRO
In addition to accepting knot presentations in a variety of formats (see usage below), knotkit contains the following tables of knot data:
This version of knotkit has support to compute the following invariants:
Khovanov homology See: M. Khovanov, A Categorification of the Jones Polynomial, Duke Math. J. 101 (2000), 359--426, arXiv:math/9908171. D. Bar-Natan, On Khovanov's Categorification of the Jones Polynomial, Algebraic and Geometric Topology 2 (2002), 337--370.
Szabo's geometric spectral sequence Computing gss was the original motivation for writing knotkit. For more information on gss, see: Z. Szabo, A geometric spectral sequence in Khovanov homology, arXiv:1010.4252, and C. Seed, Computations of Szabo's Geometric Spectral Sequence in Khovanov Homology, arXiv:1110.0735.
The Batson-Seed link splitting spectral sequence See: J. Batson, C. Seed, A Link Splitting Spectral Sequence in Khovanov Homology.
The Lipshitz-Sarkar Steenrod square on Khovanov homology See: R. Lipshitz, S. Sarkar, A Khovanov Homotopy Type, arXiv:1112.3932. R. Lipshitz, S. Sarkar, A Steenrod Square on Khovanov Homology, arXiv:1204.5776.
Bar-Natan's analogue of Lee homology See: E. S. Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005), 2, 554–586, arXiv:math/0210213. D. Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443–1499, arXiv:math/0410495. R. Lipshitz, S. Sarkar, A refinement of Rasmussen's s-invariant, arXiv:1206.3532.
The s-invariant coming from Bar-Natan's analogue of Lee homology In addition to the above references, see: J. Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 2 (2010), 419--447, arXiv:math/0402131. R. Lipshitz, S. Sarkar, A refinement of Rasmussen's s-invariant, arXiv:1206.3532.
To build knotkit on OS X, you will need the latest version of XCode (available for free in the App Store) and the GNU multiple precision arithmetic library (GMP). You can get GMP here:
knotkit also builds under Linux. In addition to GMP, you will need a C++ compiler which supports C++11. I use LLVM clang, but knotkit should build with a recent version of GCC.
knotkit doesn't have a sophisticated build system. To build knotkit, just run:
make
from the knotkit source directory. This should build the executable "kk". For instructions on invoking kk, see usage below. If you run into a problem, please contact me: Cotton Seed (cotton@math.princeton.edu).
The usage message for kk is given below. This can also be obtained by running "kk -h".
A note about output. The output for commands kh, gss, ls and lee is a .tex file which renders the bigraded homology group or spectral sequence. The output for the command sq2 matches the output for the program written by Lipshitz-Sarkar and is suitable for loading into Sage. The command s outputs a single line of text.
usage: kk