This is a follow on from my post on grid diagrams for knots you can see here. With these grid diagrams you can define a homology theory.
To define a homology theory, you need to define what are meant by chains in your space. These chains have a grading, or dimension, and there are chain maps or differentials between them. After these chains and maps have been defined, the homology groups follow and certain things need to be satisfied for these groups to represent a homology theory.
We will focus on what the chain maps and differentials are, as these are the building blocks for the whole story, and they include the best pictures! We will also consider the simpler case of the tilde version.
The chain complex of an is the free abelian group with generators that look like sets of n points (n-sets) satisfying:
- the points all lie on intersection points of the grid
- there are no points on the rightmost vertical line or top horizontal line
- there is one point on every other horizontal line
- there is one point on every other vertical line
So again, finding these n-sets of points is like doing a rubbish sudoku.
Here is a grid diagram with grid type 5. We don’t care about what the X’s and O’s are, just if they are present, and so the boxes in the grid have been shaded yellow if they feature an X or an O.
Here are 2 sets of n points we can place on the grid: the first one in red and the second one in purple. Notice they both have 5 points:
Now we need to define a differential on the chain complex. It is enough to define it on the generators. If we have 2 different n-sets x and y which are the same apart from 4 points (2 points of x differ from two points of y) let Rect(x,y) be the set of rectangles with these four points as corners, including rectangles that go off the right edge and appear at the left, or off the top edge appearing at the bottom.
Here is a choice of x and y which differ at only 4 points:
and here are two rectangles we can draw which have those 4 points as corners.
Notice for the larger (pink and green) rectangle, it has the correct points as corners only after it has disappeared from the right, to reappear on the left, and disappeared from the bottom, to reappear at the top.
We actually want to consider the set of these rectangles which contain no Xs and Os, so let X(r) be the number of Xs in a rectangle, and O(r) be the number of Os. Then we want X(r)=O(r)=0. The differential of an n-set x is then the sum of all the y‘s for which there is a rectangle with no Xs and Os, as shown below:
So for our example, and for the chosen y, this (pink) rectangle:
has X(r)=O(r)=0, and so y is included as a term in the differential for x.
So now we have defined the interesting part of the homology theory!
Finally I would like to apologise for being a week late in writing this, it’s been hectic this past fortnight 🙂