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WHAT IS THIS?

Gridlink is a little tool for manipulating the rectangular link diagrams used by Ivan Dynnikov in his paper:

Ivan Dynnikov, Recognition algorithms in knot theory. (Russian) Uspekhi Mat. Nauk 58 (2003), no. 6(354), 45--92; translation in Russian Math. Surveys 58 (2003), no. 6, 1093--1139.

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.

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 is like a castling move involving the top and bottom (left and right) horizontal (vertical) segments, thought of as lying on the 2-sphere. 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.

HOW DO YOU USE IT?

The easiest way to run the program is to type
python gridlink.py

from the directory containing the files gridlink.py and gridlink_data.py. This will start the GUI.

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 window with an image of the figure 8 knot is presented. Each link opens in a new window, and these are tracked in the Windows menu. If all of the link windows are closed the figure 8 window will reappear. Closing the figure 8 window terminates the program.

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 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 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, 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. (Question: Can any link diagram be reduced to the minimal number of segments without stabilizing?)

The "View" menu gives several choices for presenting the grid projection. You can save or load a projection together with a list of moves with "File->Save as ...", or "File->Open". You can save a Postscript rendering of the link with "File->Snapshot ...".

A link can be entered as a closed braid from the menu File->New->ClosedBraid. A dialog window 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.

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 consists of 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 column indices of the X's and the column indices of the O's.

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')