PROJECT 2: GRID DIAGRAMS AND GRID MOVES
1. Part I: Required
A grid diagram is a square grid with n × n squares some of which contain either an X or an O
in such a way that every row and every column contain exactly one X and exactly one O. Such
a diagram can be associated to a projection of an oriented link by connecting each X to the O in
the same column (oriented in that direction), and each O to the X in the same row (oriented in
that direction) so that the vertical segments pass over the horizontal ones. An example is shown
in Figure 1.
Figure 1. A grid diagram for the figure-8 knot
Problem 1. Show that every knot can be realized by a grid diagram. In other words, explain why
every knot has a grid diagram.
Similar to the Reidemeister moves, there are elementary grid moves. In other words, two grid
diagrams represent the same (topological) knot if and only if they are related by a finite sequence
of the following moves: (i) cyclic permutation of rows or columns (Figure 2), (ii) stabilization or
destabilization (Figure 3), and (iii) interchanging adjacent rows or columns if the X’s and O’s do
not interleave (Figure 4).
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Figure 2. Cyclic permutation of a row and of a column
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O
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Figure 3. Two examples of stabilization and destabilization
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Figure 4. Two examples interchanging adjacent columns
Problem 2. Give an example of a sequence of elementary grid moves that exhibits a Reidemeister
I move.
Problem 3. Give an example of a sequence of elementary grid moves that exhibits a Reidemeister
II move.
Problem 4. Give an example of a sequence of elementary grid moves that exhibits a Reidemeister
III move.
2. Part II: Extra Credit
Recall the delta move shown in Figure 5.
Figure 5. The delta move
Problem 5. Create a grid version of the delta move.
The delta move is an example of an unknotting move. That is, any diagram for a knot can be
transformed to a diagram for the unknot by a finite sequence of Reidemeister moves and delta moves.
This was shown by Hitoshi Murakami and Yasutaka Nakanishi in 1989 in a paper entitled “On a
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certain move generating link-homology” which appeared in the journal Mathematische Annalen.
Looking at their proof may help you with the following problem.
Problem 6. Show that any grid diagram can be transformed to a diagram for the unknot by a finite
sequence of elementary grid moves and the delta grid move that you defined.