Augmentations and Rulings of Legendrian Links in
Abstract.
Given a Legendrian link in , we extend the definition of a normal ruling from given by Lavrov and Rutherford and show that the existence of an augmentation to any field of the ChekanovEliashberg differential graded algebra over is equivalent to the existence of a normal ruling of the front diagram. For Legendrian knots, we also show that any even graded augmentation must send to . We use the correspondence to give nonvanishing results for the symplectic homology of certain Weinstein manifolds. We show a similar correspondence for the related case of Legendrian links in , the solid torus.
1. Introduction
Augmentations and normal rulings are important tools in the study of Legendrian knot theory, especially in the study of Legendrian knots in . Here, augmentations are augmentations of the ChekanovEliashberg differential graded algebra introduced by Chekanov in [Chekanov] and Eliashberg in [EliashbergInvariants]. Chekanov describes the noncommutative differential graded algebra (DGA) over associated to a Lagrangian diagram of a Legendrian link in combinatorially: The DGA is generated by crossings of the link; the differential is determined by a count of immersed polygons whose corners lie at crossings of the link and whose edges lie on the link. This is called the ChekanovEliashberg DGA and Chekanov showed that the homology of this DGA is invariant under Legendrian isotopy. Etnyre, Ng, and Sabloff defined a lift of the ChekanovEliashberg DGA to a DGA over in [EtnyreInvariants]. Following ideas introduced by Eliashberg in [EliashbergWave], Fuchs [FuchsAug] and ChekanovPushkar [ChekanovFronts] gave invariants of Legendrian knots in using generating families, functions whose critical values generate front diagrams of Legendrian knots, by decomposing the generating families. These are generally called “normal rulings.”
These two invariants are very closely related; Fuchs [FuchsAug], FuchsIshkhanov [FuchsIshkhanov], and Sabloff [SabloffAug] showed that the existence of a normal ruling is equivalent to the existence of an augmentation to of the ChekanovEliashberg DGA for Legendrian knots in . Here, given a unital ring , an augmentation is a ring map such that and . One of the main results of [Leverson] is that the equivalence remains true when one looks at augmentations to a field of the lift of the ChekanovEliashberg DGA from [EtnyreInvariants] to the DGA over for Legendrian knots in . We extend the result to Legendrian links in to prove the main result of this paper.
Theorem 1.1.
Let be an component Legendrian link in . Given a field , the ChekanovEliashberg DGA over has a graded augmentation if and only if a front diagram of has a graded normal ruling. Furthermore, if is even, then .
The final statement tells us that for all even graded augmentations , . In particular, if is a knot, then any even graded augmentation sends to .
For , an analogous correspondence can be shown for Legendrian links in . A Legendrian link in with the standard contact structure is an embedding which is everywhere tangent to the contact planes. We will think of them as Gompf does in [GompfHandlebody]. For an example, see Figure 2. In this paper, we extend the definition of normal ruling of a Legendrian link in to a Legendrian link in . We can then define the ruling polynomial for a Legendrian link in and show that the ruling polynomial is invariant under Legendrian isotopy.
Theorem 1.2.
The graded ruling polynomial with respect to the Maslov potential (which changes under Legendrian isotopy) is a Legendrian isotopy invariant.
In [Ekholms1s2], Ekholm and Ng extend the definition of the ChekanovEliashberg DGA over to Legendrian links in . The main result of this paper uses Theorem 1.1 to extend the correspondence between normal rulings and augmentations to a correspondence for Legendrian links in .
Theorem 1.3.
Let be an component Legendrian link in for some . Given a field , the ChekanovEliashberg DGA over has a graded augmentation if and only if a front diagram of has a graded normal ruling. Furthermore, if is even, then .
Notice that one can consider Legendrian links in as being Legendrian links in . In this way, this result is a generalization of the correspondence in [Leverson] and Theorem 1.1.
Along with the work of Bourgeois, Ekholm, and Eliashberg in [BEE], Theorem 1.3 gives nonvanishing results for Weinstein (Stein) manifolds. In particular:
Corollary 1.4.
If is the Weinstein manifold that results from attaching handles along a Legendrian link to and has a graded normal ruling, then the full symplectic homology is nonzero.
This follows from Theorem 1.3 as the existence of a normal ruling implies the existence of an augmentation to , which, by [BEE], is necessary for the full symplectic homology to be nonzero.
We show a correspondence for Legendrian links in the jet space of the circle . In [NgSolidTorus], Ng and Traynor extend the definition of the ChekanovEliashberg DGA to Legendrian links in . In [LavrovSolidTorus], Lavrov and Rutherford extend the definition of normal ruling to a “generalized normal ruling” of Legendrian links in and show that the existence of a generalized normal ruling is equivalent to the existence of an augmentation to of the ChekanovEliashberg DGA of a Legendrian link in . In §LABEL:sec:solidTorus, we show that this correspondence holds for augmentations to any field of the ChekanovEliashberg DGA over .
Theorem 1.5.
Let be a Legendrian link in . Given a field , the ChekanovEliashberg DGA over has a graded augmentation if and only if a front diagram of has a graded generalized normal ruling.
1.1. Outline of the article
In §2 we recall background on Legendrian links in and . We give definitions of the ChekanovEliashberg DGA over , with sign conventions, and augmentations of the DGA in both and . We also define normal rulings for links in and show that the ruling polynomial is invariant under Legendrian isotopy. In §3, we prove Theorem 1.1. In §4, given an augmentation, we construct a normal ruling proving one direction of Theorem 1.3. In §LABEL:sec:rulingToAug, given a normal ruling, we construct an augmentation, finishing the proof of Theorem 1.3. In §LABEL:sec:solidTorus, we prove Theorem 1.5. In the Appendix, we give the nonvanishing symplectic homology result.
1.2. Acknowledgements
The author thanks Lenhard Ng and Dan Rutherford for many helpful discussions. This work was partially supported by NSF grants DMS0846346 and DMS1406371.
2. Background Material
2.1. Legendrian Links in
In this section we will briefly discuss necessary concepts of Legendrian links in . We will follow the notation in [Ekholms1s2].
Definition 2.1.
Let . A tangle in is Legendrian if it is everywhere tangent to the standard contact structure . Informally, a Legendrian tangle in is in normal form if

meets and in groups of strands, where the groups are of size , from top to bottom in both the and projections,

and within the th group, we label the strands by from top to bottom at in both the and projections and in the projection, and from bottom to top at in the projection.
Every Legendrian tangle in normal form gives a Legendrian link in by attaching handles which join parts of the projection of the tangle at to the parts at . In particular, the th handle joins the th group at to the th group at and connects the strands in this group with the same label at and through the handle. See Figure 2.
Every Legendrian link in has an diagram of the form given by Gompf in [GompfHandlebody], which we will call Gompf standard form. The left diagram of Figure 2 is an example of a link in Gompf standard form. Any link in Gompf standard form can be isotoped to a link whose projection is obtained from the diagram by resolution. The resolution of an diagram of a link is obtained by the replacements given in Figure 1. For an example, see Figure 2. By [Ekholms1s2], an diagram obtained by the resolution of an diagram of a link in Gompf standard form is in normal form. Thus, we can assume that the diagram of any Legendrian link is in normal form.
2.2. Definition of the DGA and augmentations in
This section contains an overview of the differential graded algebra over presented by Ekholm, Ng in [Ekholms1s2]. Let be a Legendrian link in , where the denote the components of and . Let be the number of strands of which go through the th handle with the total number of strands at .
2.3. Internal DGA
We will define the internal DGA for a Legendrian link in , but one can easily extend the definition to the internal DGA for a Legendrian link in by defining the internal DGA as follows for each handle separately.
Let be the tuple where is the rotation number of the th component and let be the tuple of a choice of Maslov potential for each strand passing through the handle (see §2.5).
Let denote the DGA defined as follows. Let be the tensor algebra over generated by for and for and . Set , , and
for all . Define the differential on the generators by
where , is the Kronecker delta function, and we set for . Extend to by the Leibniz rule
From [Ekholms1s2], we know has degree , , and is infinitely generated as an algebra, but is a filtered DGA, where is a generator of the th component of the filtration if .
Given a Legendrian link , we can associate a DGA to each of the handles. We then call the DGA generated by the collection of generators of for with differential induced by , the internal DGA of .
2.4. Algebra
Suppose we have a Legendrian link in normal form with exactly one point labeled within the tangle (away from crossings) on each link component of (corresponding to ). We will discuss the case where there is more than one base point on a given component in §2.11.
Notation 2.2.
Let denote the crossings of the tangle diagram in normal form. Label the handles in the diagram by from top to bottom. Recall that denotes the number of strands of the tangle going through the th handle. For each , label the strands going through the th handle on the left side of the diagram from top to bottom and from bottom to top on the right side, as in Figure 2.
Let be the tensor algebra over generated by

;

for and ;

for , , and .
(In general, we will drop the index when the handle is clear.)
2.5. Grading
The following are a few preliminary definitions which will allow us to define the grading on the generators of .
Definition 2.3.
A path in is a path that traverses part (or all) of which is connected except for where it enters a handle, meaning, where it approaches (respectively ) along a labeled strand and exits the handle along the strand with the same label from (respectively ). Note that the tangent vector in to the path varies continuously as we traverse a path as the strands entering and exiting handles are horizontal.
The rotation number of a path is the number of counterclockwise revolutions around made by the tangent vector to as we transverse . Generally this will be a real number, but will be an integer if and only if is smooth and closed.
Thus, the rotation number is the rotation number of the path in which begins at the base point on the link component and traverses the link component, following the orientation of the component. In the case where is a link with components , we define
Define
If is the resolution of an diagram of an component link in Gompf standard form, then the method assigning gradings follows: Choose a Maslov potential that associates an integer modulo to each strand in the tangle associated to , minus cusps and base points, such that the following conditions hold:

for all and all , the strand labeled going through the th handle at and the must have the same Maslov potential;

if a strand is oriented to the right, meaning it enters the handle at and exits at , then the Maslov potential of the strand must be even. Otherwise the Maslov potential of the strand must be odd;

at a cusp, the upper strand (strand with higher coordinate) has Maslov potential one more than the lower strand.
The Maslov potential is welldefined up to an overall shift by an even integer for knots. (In [Ekholms1s2], Ekholm and Ng give another method for defining the gradings using the rotation numbers of specified paths.)
Set and , where means the Maslov potential of the strand with label going through the th handle. It remains to define the grading on crossings in the tangle, crossings resulting from resolving right cusps, and crossings from the halftwists in the resolution. If is crossing of tangle , then let
where is the strand which crosses over the strand at in the projection of . If is a right cusp, define (assuming there is not a base point in the loop). If is a crossing in one of the halftwists in the resolution where strand crosses over strand (), then
2.6. Differential
It suffices to define the differential on generators and extend by the Leibniz rule. Define . Set on as in §2.3.
In [Ekholms1s2], the DGA on crossings is defined by looking for immersed disks in the diagrams of Legendrian links, (see the left diagram in Figure 3). However, Ekholm and Ng note that it is equivalent to look for immersed disks in dip versions of the diagram, (see the right diagram in Figure 3). See Figure 4 for the labeling of the crossings in Figure 3.
Definition 2.4.
Let be generators. Define to be the set of orientationpreserving immersions
(up to smooth reparametrization) that map to the dip version of such that

is a smooth immersion except at ,

are encountered as one traverses counterclockwise,

near , covers exactly one quadrant, specifically, a quadrant with positive Reed sign near and a quadrant with negative Reeb sign near , where the Reeb sign of a quadrant near a crossing is defined as in Figure 5.
To each immersed disk, we can assign a word in by starting with the first corner where the quadrant covered has negative Reeb sign, , and listing the crossing labels of all negative corners as encountered while following the boundary of the immersed polygon counterclockwise, . We associate an orientation sign to each quadrant in the neighborhood of a crossing , defined in Figure 5, and use these to define the sign of a disk to be the product of the orientation signs over all the corners of the disk. We denote this sign by . In many cases there is a unique disk with positive corner at (with respect to Reeb sign) and negative corners at and in these we define to be the sign of the unique disk. (In exceptional cases there may be more than one disk with positive corner at and negative corners at .)
Define or to be the signed count of the number of times one encounters the base point while following counterclockwise, where the sign is positive if we encounter the base point while following the orientation of the link component and negative if we encounter the base point while going against the orientation.
We define
and extend to by the Leibniz rule.
In [Ekholms1s2], Ekholm and Ng prove the map has degree and is a differential, .
Example 2.5.
The following is the definition of the DGA for the Legendrian link in Figure 6. Here is generated by over . We set for . Define a Maslov potential on the strands near by
Then we have the following gradings: , , ,
where is the crossing of the strands in the bottom handle. Since , we know for .
For ease of notation, we will use to denote . We then have the following differentials: