knotkit/README

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

1. INTRO
2. BUILDING
3. USAGE
4. UPCOMING CHANGES
5. FOR DEVELOPERS

1. INTRO

In addition to accepting knot presentations in a variety of formats
(see usage below), knotkit contains the following tables of knot data:

* The Rolfsen knot tables through 10 crossings, extracted from
  Bar-Natan's Mathematica package, KnotTheory`,
* The Hoste-Weeks-Thistlewaite knot tables through 16 crossings taken
  from knotscape, and
* and the Morwen Thistlewaite hyperbolic link tables taken from
  SnapPy.

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.

2. BUILDING

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:

http://gmplib.org/

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).

3. USAGE

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 <invariant> [options...] <link>
  compute <invariant> for knot or link <link>
<invariant> can be one of:
  kh: Khovanov homology
  gss: Szabo's geometric spectral sequence
  lsss: Batson-Seed link splitting spectral sequence
    component weights are 0, 1, ..., m
  sq2: Lipshitz-Sarkar Steenrod square on Z/2 Kh
    output suitable for Sage
  leess: spectral sequence coming from Bar-Natan analogue of Lee's
    deformation of Khovanov's complex (whew!)
  s: Rasmussen's s-invariant coming from lee
output:
    kh, gss, lsss, leess: .tex file
    sq2: text in Sage format
    s: text
options:
  -r         : compute reduced theory
  -h         : print this message
  -o <file>  : write output to <file>
                (stdout is the default)
  -f <field> : ground field (if applicable)
                (Z2 is the default)
  -v         : verbose: report progress as the computation proceeds
<field> can be one of:
  Z2, Z3, Q
<link> can be one of:
  - the unknot, e.g. U or unknot
  - a torus knot, e.g. T(2,3)
  - a Rolfsen table knot, e.g. 10_124
  - a Hoste-Thistlethwaite-Weeks knot, e.g. 11a12 or 12n214
  - a Morwen Thistlethwaite link, e.g. L8n9 or L13n8862
  - a planar diagram, e.g.
      PD[X[1, 4, 2, 5], X[3, 6, 4, 1], X[5, 2, 6, 3]] or
      PD[[1, 4, 2, 5], [3, 6, 4, 1], [5, 2, 6, 3]]
  - a Dowker-Thistlethwaite code, e.g.
      DTCode[6,8,2,4],
      DT[dadbcda] or
      DT[{6, -8}, {-10, 12, -14, 2, -4}]
  - a braid, e.g. BR[2, {-1, -1, -1}]
  - disjoint union (juxtaposition), e.g. T(2,3) U

4. UPCOMING CHANGES

The following changes are currently planned:

* support for Z/p, p arbitrary prime and F(x) field of rational functions
* Roberts' totally twisted Khovanov homology
* the E^3 page of the twisted spectral sequence Kh(L) => \widehat{HF}(\Sigma_L)
* spectral sequences of PIDs: lsss over F[x], gss over Z
* maps induced by cobordisms

5. FOR DEVELOPERS

If you are interested in contributing to knotkit or using it for a new
application, please contact me: Cotton Seed
(cotton@math.princeton.edu).  I am interested in developing knotkit to
be an open platform for performing computations arising in
low-dimensional topology.