κ

kappa computes κ, an invariant of braid conjugacy classes. It can also compute various reduced flavors. For information about κ, check out my and Diana Hubbard's paper.

Here's the latest version of the source code: source

kappa is written in Haskell, for better or worse. The source code and documentation will soon be available at Hackage, the central repository for Haskell libraries. You can install by running cabal install kappa on a machine with Cabal.

If you just want to run the program on your favorite braid, you can download binaries here: Windows, Mac (coming soon!). The program can be run from the commandline. The syntax is

computekappa MARK WORD WIDTH

The WORD \sigma_2\sigma_3\sigma_1^{-1}\sigma_2^{-2} in standard Artin generators should be entered as [2,3,-1,-2,-2]. The WIDTH is the braid index, i.e. the number of strands. If MARK is positive, the program will compute \kappa in the reduced Khovanov subcomplex with a basepoint on the arc numbered MARK. If the MARK is negative, the program instead uses the reduced Khovanov quotient complex. If the MARK is 0, the program uses unreduced Khovanov homology. All computations are done with Z/2Z coefficients.

For example, to compute κ for a two-strand trefoil: computekappa 0 [1,1,1] 2

I am indebted to John Baldwin, Malte Milatz, and Dylan Thurston for sharing their code for related projects.