This is part II of a post on the relationship between augmentations and rulings. Here is part I if you missed it.
Given a front diagram of a Legendrian knot in plat position, we will use Sabloff’s method of dips to show that if there exists an augmentation of the Chekanov-Eliashberg DGA, then there exists a normal ruling of the diagram. We will extend the augmentation of the diagram to an augmentation of the dipped diagram satisfying Property (R) which will simultaneously build a normal ruling of the original front diagram.
We will extend to by adding dips to the diagram from left to right and extending the augmentation and ruling as we go as follows: If a crossing is augmented in the original front diagram, then augment in the dipped diagram. In a dip, we decide whether a crossing in the -lattice is augmented by following Property (R).
Property (R): At the dip between and , a crossing in the -lattice is augmented iff the two strands which cross there are paired in the ruling of the original front diagram between and .
A crossing in the -lattice between crossings and is augmented based on whether the strands which cross at are crossing or companion strands in the ruling of the original front diagram and which diagram below looks like .
(Figure from Sabloff’s paper.)
In particular, if has configuration ____ and is augmented, then is augmented if the strands crossing at are _____ strands. (Use the following to fill in this statement.)
(b) or (c), crossing and companion
(e) or (f), companion
Note that this definition relies on simultaneously building a ruling for the original front diagram. So, let us describe how to build the ruling. If we look at the left end of the ruling of a front diagram in plat position, the ruling will have strand paired with strand for all . To extend the ruling over a crossing , if is augmented and it looks like configuration (a), (b), or (c) just to the left of , then switch the ruling at . Otherwise, don’t switch the ruling.
One can check that this gives us an augmentation of the dipped diagram. For example, suppose is augmented and has configuration (c) just to the left. We will denote augmented crossings by a large dot. Suppose strands and cross at and that strand is paired with strand and strand is paired with strand between crossings and .
Since is augmented and looks like configuration (c) to the left, our ruling is switched at .
(I apologize for the crude drawings, but Inkscape wasn’t cooperating.)
Now let us use this diagram to compute . Recall that the only disks that contribute to are totally augmented disks, disks with a positive corner at and with all negative corners augmented by . Each such disk will contribute 1 to , so as long as there are a even number of such disks, .
There are no such disks for , so . A more interesting computation is the crossing in the -lattice of strands and . We see that there are two such disks.
We also have the following disks for the crossing in the -lattice of strands and . So .
A crossing with more standard disks appearing, is the crossing in the $b$-lattice where strands and cross. It has the following disks.
There are no such disks for any crossing in the -lattice, so on the -lattices. One can check that for all remaining crossings.
While the dipped diagram makes differentials much easier, they tend to be a pain to deal with. Luckily, there’s a shortcut for finding the ruling associated to an augmentation of the original front diagram, without going through the dipped diagram:
We will modify our augmentation of the crossings of the plat position diagram while extending the ruling from left to right. Start the ruling at the left cusps like usual, and extend as follows. If a crossing looks like configuration (a), (b), or (c) to the left and the crossing is augmented, then switch the ruling at the crossing and update the augmentation on the crossings to the right based on the number of disks to the right which contribute and involve the crossing strands.
Otherwise do not switch at the crossing. However, if a crossing looks like configuration (e), or (f) to the left and is augmented, then update the augmentation of the crossings to the right based on the number of disks to the right which contribute and involve the companion strands.
This is easier to see in an example. Let’s look at the trefoil where all crossings of degree are augmented.
It turns out the augmentation where only and are augmented will give the same ruling. Ng and Sabloff’s paper The Correspondence Between Augmentations and Rulings for Legendrian Knots discusses how this is a many-to-one map and gives the number of -graded augmentations going to the same -graded ruling for and or odd. (You’ll have to read up on what a -graded augmentation is in the paper.)