Gridlink is a tool for manipulating the rectangular link diagrams which were introduced as variant forms of arc presentations in
Peter R. Cromwell, "Embedding knots and links in an open book I: Basic properties," Topology Appl. 64 (1995), 37--58.
These diagrams were later used by Ivan Dynnikov in his paper:Ivan Dynnikov, "Arc-presentations of links. Monotonic simplification," arXiv:math.GT/0208153.
Dynnikov proved in the paper above that any rectangular diagram of an unknot or split link can be monotonically reduced to a trivial diagram by a sequence of elementary moves. Thus this gives a recognition algorithm for grid diagrams of the unknot.
More recently (Fall 2006) these diagrams have been used to give a
combinatorial description of the knot and link Floer homology.
Ciprian Manolescu, Peter Ozsvath and Sucharit Sarkar, "A combinatorial description of knot Floer homology," arXiv:math.GT/0607691.
Ciprian Manolescu, Peter Ozsvath, Zoltan Szabo and Dylan Thurston, "On combinatorial link Floer homology," arXiv:math.GT/0610559.
A rectangular link diagram (or gridlink) is a polygonal link projection consisting of horizontal and vertical line segments lying in the integer grid, such that whenever two segments cross, the vertical segment is the overcrossing. The diagram is also required to be in standard position, in the sense that no two segments are colinear.
Following Cromwell, Dynnikov defines three elementary moves on a
rectangular diagram: the castling, or exchange move; the cyclic
permutation, and the destabilization move. The castling move is an
isotopy which drags one vertical (horizontal) segment across an
adjacent one, interchanging the x (y) coordinates of the two segments,
and fixes all segments which do not meet the two that are being
interchanged. (Because it is an isotopy, a castling move can only be
performed when the projections of boundary 0-spheres of the two
segments are unlinked.) The cyclic permutation move is an isotopy in
which the horizontal (vertical) segment at the top or bottom (left or
right) is moved under (over) the diagram to the opposite side. The
simplest destabilization move removes a pair of consecutive segments
of length 1. However, the boundary of a segment of length 1 can never
be linked. So there is a more general destabilization move that
removes any segment of length 1; this is the destabilization
implemented by the program. There are four types of
stabilization/destabilization moves, and the user can restrict which
these types will be allowed.
After completing the installation, the program can be run from the
command line by just typing gridlink . It is also
possible to run it by typing python gridlink.py from
the directory containing the files gridlink.py and gridlink_data.py.
Double click the figure 8 icon to start up the Macintosh standalone application.
If you want to have access to the internal objects you can also load
gridlink as a python module from the python interpreter with the
command:
>>> import * from gridlink
For this to work, make sure that your python interpreter can find the
two files gridlink.py and gridlink_data.py. This can be done by
starting the python interpreter from the directory containing the
files, or by putting them in a directory that is specified in your
PYTHONPATH variable, or by putting them in the python site-packages
directory.
When the program starts, a small "app window" with an image of the figure 8 knot opens. Each link will open in a new window, and these windows will be tracked in the "Windows" menu. The app window disappears as soon as a link is opened. If all of the link windows are closed the app window will reappear. Closing the app window stops the program, as does the "File->Exit" menu.
Inside a link window, when two segments are highlighted in blue, click the mouse to exchange. Click and drag the mouse on the background to do cyclic permutation. Gridlinks are oriented, and the orientation is displayed with an arrow head on each vertical segment. (They can also be displayed as a matrix of dots, or X's and O's.) The orientation of a component can be reversed by clicking the "reverse" button and then selecting the component with the mouse. The "reflect" botton rotates the diagram by 90 degrees; due to the crossing convention this is equivalent to reversing all crossings. The buttons labelled "NE", "NW" "SE", "SW" select one of the four types of stabilization moves. (The checkboxes under these buttons can be used to limit which types of stabilizations/destabilizations are allowed.) After clicking a stabilization button, select a segment to be stabilized. If a vertical segment is selected, the move takes place at the tail of the segment. To perform a move at the head of a vertical segment, select the following horizontal segment. In this case, the stabilization move is equivalent to stabilizing at the tail of the vertical segment and then transporting the new length 1 segment to the head by exchange moves. To destabilize, click the "destab" button and select a length 1 segment to remove.
There are shortcut keys for all operations except for stabilization: d to destabilize, r to reverse, f to reflect, u to undo, and arrow keys for cyclic permutations.
The program records all moves, and the list can be displayed by running the "print_moves" method at the python prompt. The number of moves is displayed at the lower right corner of the gridlink window. The "Moves->Reset" menu action clears the list of saved moves and restarts the counter. The "Moves->Review" menu action allows you to step forward or backward through your sequence of moves, displaying the projection at each step. The "Moves->Simplify" menu action applies a random sequence of 10000 (unrecorded) elementary moves, performing (allowed) destabilizations when possible. In most cases this will reduce the presentation to something close to minimal. If not, try again. Note that the process does not involve stabilization. In contrast with Dynnikov's result for projections of the trivial knot, Lenny Ng has pointed out to us that the the (2,3) cable of the (2,3) torus knot has a grid projection which cannot be minimized without stabilization. He provided the following XO-description of this projection:
X = {1,0,7,14,13,3,2,8,15,5,4,10,9,16,6,12,11}
O = {10,9,16,0,6,12,11,1,7,14,13,3,2,8,15,5,4}
This example is discussed in the appendix to
W. Menasco, "An addendum on iterated torus knots"
arXiv:math.GT/0610566.
The "File" menu allows a projection to be saved, loaded, printed or created. Save or load a projection together with a list of moves with "File->Save as ...", or "File->Open". Write a Postscript rendering of the link with "File->Snapshot ...". The "File->New" menu is used to create a projection, as described below. The "File->Exit" menu terminates the program.
The "View" menu gives several options for the graphical represention of the projection and display of addition data about the projection, such as winding numbers and the Thurston-Bennequin number.
The "Moves" menu allows the user to step backwords and forwards through the sequence of saved moves.
The "Invariants" menu allows computation of the Heegaard Floer Homology groups with mod 2 coefficents (for knots). Gridlink has an embedded version of the software written by John A. Baldwin and W. D. Gillam.
The "Windows" menu keeps track of all link projection windows that are open.
A link can be entered as a closed braid from the menu File->New->ClosedBraid. A dialog box will ask for the number of strands and a word in the braid group. For example, the word 1,1,-2,1,3,-2,3,4,-3,4 describes a 5 strand braid representation of knot 9_12.
The program also contains a table of closed braid representions of knots up to 12 crossings, generated from a query of Chuck Livingston's knotinfo database: http://www.indiana.edu/~knotinfo/ To load a knot from the table use the File->New->Knot menu. Some examples of valid knot names are: 9_12, 11a_123, 12n_123.
Finally, the menu File->New->XO link allows entry of a gridlink as a matrix containing exactly one X and one O in each row and column. The link is constructed as a vertical segment from the X to the O in each column, and a horizontal segment from the O to the X in each row. Enter the link by listing the row indices of the X's and the row indices of the O's. NOTE: rows are numbered from bottom to top, columns from left to right. Indices begin at 0.
Example session (using the interpreter):
% python
>>> from gridlink import *
>>> A = GridlinkApp()
>>> #Here are a couple of links that cannot be reduced by elementary moves.
>>> #Thanks to Dylan Thurston for these.
>>> L4a1 = XOlink(A, [1,2,5,0,3,4], [3,0,1,4,5,2])
>>> Whitehead = XOlink(A, [6,3,2,1,4,5,0], [2,1,0,5,6,3,4])
>>> #Here are a couple of interesting unknots. Thanks to Ian Agol for these.
>>> K== XOlink(A, [5,1,7,3,12,4,6,0,11,2,8,13,19,10,21,15,17,9,18,14,20,16],
[0,4,2,6,5,8,1,7,3,10,12,9,11,18,14,20,13,16,15,19,17,21])
>>> K = XOlink(A, [7,14,15,12,13,9,10,2,1,0,4,3,8,6,5,11],
[15,10,13,14,11,12,8,9,3,2,1,5,4,0,7,6])
>>> K9_12 = ClosedBraid(A,5,[1,1,-2,1,3,-2,3,4,-3,4])
>>> K = Knot(A,'11a_123')