Now we’ll work on going from rulings to augmentations. Fuchs does this using what he calls “splashes” in diagrams, but I find it easier to see this using Sabloff’s method of defining an augmentation for the dipped diagram as one can then get an augmentation for the original diagram.
Given a ruling for the original diagram in plat position, we will define an augmentation for the dipped diagram. First, augment if the ruling is switched at , augment , the crossing of strands and in the -lattice, if strands and are paired between and , (this is what we called Property (R) in part II), and augment , the crossing of strands and in the -lattice, if one of the following holds:
- ruling looks like (a) at the previous crossing and strands and are crossing strands,
- ruling looks like (b) or (c) at the previous crossing and strands and are crossing or companion strands,
- ruling looks like (e) or (f) at the previous crossing and strands and are companion strands.
Recall the various crossing configurations.
Let’s check for a couple of these cases that this gives an augmentation of the dipped diagram. In other words, check that for each crossing in the dipped diagram there are an even number of totally augmented disks in the diagram with positive corner at that crossing.
First, check the left end of the diagram. Since, in the ruling strands and are paired at the left, we know the crossing in the first -lattice of strands and is augmented for .
Most of the crossings in the dips I will leave for you to check, but we will check the dip after a crossing of configuration (c). Thus the ruling is switched at that crossing. Suppose strands and cross at the crossing and that strand is paired with and strand is paired with .
Since the ruling is switched at the crossing, we know the crossing is augmented. We also see that the following other crossings are augmented as well, from the pairing of the strands in the ruling.
To check whether on the crossings in the dip after the original crossing, look for totally augmented disks.
Clearly there aren’t any totally augmented disks with positive corner at , so .
We see that there are two totally augmented disks contributing to of the crossing in the -lattice of strands and and so . (Recall that we are working mod 2.)
We see that there are two totally augmented disks contributing to of the crossing in the -lattice of strands and and so .
Similarly, we have disks for crossings and in the -lattice of, respectively, strands and and strands and .
None of the remaining crossings in the – or -lattice have totally augmented disks, so we have checked that on this region of the dipped diagram, is an augmentation.
Now, let’s look at the right end of the dipped diagram. Since we have the ruling of the original diagram, we know that at the right end of the diagram, strands and are paired in the ruling for . Following our algorithm, this means that the crossings in the -lattice of strands and are augmented for .
Thus we have the following totally augmented disks for .
Again, we see that there are two totally augmented disks with positive corner at , so .
Thus, with some checking of the remaining cases, we have shown that the augmentation of the dipped diagram we defined, is in fact an augmentation and so, given a way to define an augmentation of the dipped diagram of a knot from a ruling of the knot.