# Setup for the Relationship Between Augmentations and Rulings

The Chekanov-Eliashberg DGA is an invariant which associates a DGA over the integers mod 2 to a Lagrangian projection of a Legendrian knot. The generators come from crossings and the differential comes from counting immersed polygons whose edges lie in the diagram of the knot and whose vertices lie at crossings. (For more on how it is defined, see Chekanov’s paper Differential Algebras of Legendrian Links.) One can then look at augmentations of the DGA which are algebra maps $\epsilon:\mathcal{A}\to\mathbb{Z}/2$ which satisfy $\epsilon\circ\partial=0$ and $\epsilon(1)=1$. If $\epsilon(c)=1$, we call $c$ augmented. If we go back to the diagram of our knot, we can define a “normal ruling” of the diagram. In the paper Chekanov-Eliashberg invariants of Legendrian knots: Existence of augmentations, Fuchs uses “splashed” diagrams to show that if a Legendrian knot has a normal ruling, then it has an augmentation. Then, in Augmentations and rulings of Legendrian knots, Sabloff uses “dipped” diagrams to show that if a knot has an augmentation, then it has a normal ruling. We will use Sabloff’s method of “dipped” diagrams to show both results. However, in this posting, I will only give the necessary notation and definitions. Sorry there are so many.

First off, a ruling of the front diagram of a knot consists of a one-to-one correspondence between the left cusps and the right cusps and, for each pair of corresponding cusps, two paths in the front diagram that join them. They must also satisfy the conditions that two paths in the ruling meet only at crossing or cusps and the interior of the two paths joining corresponding cusps are disjoint and so only meet at the cusps and so bound a topological disk. Thus, the paths in a ruling cover the front diagram. Near a crossing, we call the two ruling paths (one from each of two paired cusps) which are on strands that cross, crossing paths and the paths paired with these, companion paths. We say the ruling is switched at a crossing if one of the crossing paths lies entirely above the other near the crossing. If all of the switched crossings of a ruling are one of the following, we call the ruling normal.

(Figure from Sabloff’s paper.)

For example, the three normal rulings of the trefoil are

(Figure from Sabloff’s paper.)

Recall that the DGA for the trefoil

is $\mathbb{Z}/2\langle a_1,a_2,a_3,q_1,q_2\rangle$ with the only nontrivial differentials

$\partial q_1=1+a_1+a_3+a_3a_2a_1$
$\partial q_2=1+a_1+a_3+a_1a_2a_3$

So an example of an augmentation for the trefoil is $\epsilon(a_1)=1=\epsilon(a_2)$ and $\epsilon(a_3)=0$.

Unfortunately, we still need a few more definitions.

We will assume that the front diagram of our knot is in plat position, in other words, all the left cusps have the same x-coordinate, all the right cusps have the same x-coordinate, and none of the crossings have the same x-coordinate. The trefoil above is in plat position. We can make this assumption as a series of Legendrian versions of the Reidemeister II moves and planar isotopies will put any front diagram in plat position. Call the crossings in this diagram $c_1,\dots,c_n$.

Suppose we have $m$ right cusps and at any x-coordinate in the diagram, label the strands in the diagram from bottom to top by $1,\ldots,2m$.

A dip is constructed by a sequence of Reidemeister II moves, which looks as follows in the front projection and Lagrangian projection.

(Picture from Sabloff’s paper.)

The dipped diagram involves introducing a dip between each crossing in the plat position diagram and in between the left, respectively right, cusps and the first, respectively last, crossing. Each Reidemeister II move introduces two new crossings. In each dip, call the new crossings which appear on the left the $b$lattice and the right the $a$lattice. While dipped diagrams have many more crossings than the original knot diagram, the differential $\partial$ on $\mathbb{Z}$-differential graded algebra is generally much simpler. Note that in the dipped diagrams, the differential of crossings in a $b$-lattice involve at most $c_k$, crossings in the previous $a$-lattice, and crossings in the $a$-lattice, the differential of crossings in an $a$-lattice only involve crossings in the $b$-lattice, and the
differential of $c_k$ is $0$ for all $k$. This greatly reduces the totally augmented disks (disks contributing to the differential of the crossing where all negative corners are augmented) for which to look to compute whether we have an augmentation. (These are the only disks which contribute to $\epsilon\circ\partial$.)

This is most of the setup necessary to show both that augmentations imply rulings and rulings imply augmentations.