Positive Knots and Lagrangian Fillability

This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact $\rr^3$ and the hierarchy of positive, strongly quasi-positive, and quasi-positive knots. On one hand, results of Eliashberg and especially Boileau and Orevkov show that every Legendrian knot with an exact, embedded Lagrangian filling is quasi-positive. On the other hand, we show that if a knot type is positive, then it has a Legendrian representative with an exact embedded Lagrangian filling. Further, we produce examples that show that strong quasi-positivity and fillability are independent conditions.


Introduction
Properly embedded Lagrangian submanifolds of B 4 whose boundaries are Legendrian links in S 3 , called fillings of the Legendrian links, are of interest in a variety of fields: in smooth knot theory, Lagrangian fillings minimize the slice genus of a link [6]; in Legendrian knot theory, Lagrangian fillings induce augmentations of the Legendrian Contact Homology DGA [8,9,18]; and Lagrangian fillings can even be used to answer questions about complex algebraic curves [5]. These considerations motivate the following question: Which smooth knot types have Legendrian representatives with Lagrangian fillings? Such a smooth knot type is termed fillable. In analyzing this question, we work in the equivalent setting of Legendrian links in the standard contact R 3 and Lagrangian fillings in the symplectization R×R 3 . We further require the Lagrangian fillings to be exact, orientable, embedded, and collared, i.e. equal to R × Λ outside a compact set.
Initial progress on the question above indicates a close relationship to the hierarchy of positivity in smooth knot theory. To describe the hierarchy, let BP be the set of braid positive knot types, P be the set of positive knot types, SQP be the set of strongly quasi-positive knot types, and QP be the set of quasi-positive knot types. The following relationships are well known (see [13,21,24], for example): BP P SQP QP.
The first main result of this paper delineates a sufficient condition for a smooth knot type to be fillable:  To make progress towards a necessary condition, we begin by noting that quasi-positivity is necessary for fillability. To see this, first note that a result of Eliashberg [10] can be extended to show that a Lagrangian filling of a Legendrian knot may be perturbed to a symplectic filling of a transverse knot (see [5] for more details). From there, use the following result of Boileau and Orevkov [3]: if a smooth knot type has a transverse representative with a symplectic filling, then it is quasi-positive.
As we shall see in Section 4, however, not all quasi-positive knots are fillable. Further, the intermediate condition of strong quasi-positivity is independent of fillability: There exists a fillable knot that is quasi-positive, but not strongly quasi-positive, and there exists non-fillable knot that is strongly quasi-positive.
Since this theorem shows that strong quasi-positivity is not relevant to fillability, we must seek an alternative condition to characterize fillable knots. Based on the results above and a survey of quasi-positive knots up to ten crossings, 1 we make the following conjecture: Conjecture 1.3. A smooth knot type K is fillable if and only if it is quasipositive and the HOMFLY bound on the maximum Thurston-Bennequin number of K is sharp.
One implication of the conjecture is true: if a smooth knot is fillable, then, as discussed above, it is quasi-positive. Further, the Legendrian contact homology DGA of the fillable Legendrian representative has an augmentation [8,18], hence a graded ruling [12,23], and hence the HOMFLY bound is sharp [22].
The remainder of the paper is organized as follows: after reviewing background on the various notions of positivity touched on above, on rulings of Legendrian knots, and on the construction of Lagrangian cobordisms via handle attachment in Section 2, we prove Theorem 1.1 in Section 3 and Theorem 1.2 in Section 4.
Acknowledgements. We thank Matt Hedden for several stimulating discussions about the material in this paper. The authors were partially supported by NSF grant DMS-0909273.

Background
In this section, we review positivity and (strong) quasi-positivity of smooth knots, rulings of Legendrian knots, and various constructions of Lagrangian 1 We used [2] for a list of quasi-positive knots, Morrison's code in the KnotTheory package [1] and Morton and Short's C++ program [14,15] to compute the HOMFLY polynomials, Gridlink [7] to find Legendrian representatives of quasi-positive knots, and our own constructions of Lagrangian fillings.
fillings. We assume that the reader is familiar with basic notions of Legendrian knot theory as discussed, for example, in Etnyre's survey [11]. In particular, we assume familiarity with front diagrams, the classical invariants, and the HOMFLY bound on the Thurston-Bennequin number.
We define a (strongly) quasi-positive braid to be a product of positive (embedded) bands, and we say that an oriented link is (strongly) quasipositive if it is the closure of such a braid.
The relationship between these notions of positivity -in particular, the inclusions displayed in Equation (1.1) -has been treated extensively in the literature; see [13,21,24] for surveys.

Rulings of Legendrian Links.
A ruling is a combinatorial object associated to the front diagram of a Legendrian link. Hereafter, assume that all Legendrian links have been perturbed so that the cusps and crossings in their front diagrams have distinct x-coordinates. We define a ruling of such a front to be a one-to-one correspondence between left and right cusps, together with pairs of paths (called companions) in the front joining corresponding cusps. The companion paths must satisfy the following conditions: (1) paired paths intersect only at the cusps they join; and (2) unpaired paths intersect only at crossings. In particular, the x-coordinate of each path is strictly monotonic and the paths cover the front. Further, since the companion paths meet only at the cusps, each pair of companions bounds a ruling disk in the plane.
To refine the notion of a ruling, we refer to the paths incident to a crossing of the front diagram as crossing paths. At a crossing, either the two crossing paths pass through each other or one path lies entirely above the other. In the latter case, we say that the ruling is switched at the crossing. We call a ruling normal if, at each switched crossing, the crossing paths have one of the configurations shown in Figure 1. We can rephrase this definition in terms of ruling disks: a ruling is normal if, near a switch, the interiors of the disks involved in the switch are either nested or disjoint. If all switches occur at positive crossings, then we say that the ruling is oriented. 2    The proof of Theorem 1.1 relies on rulings that switch at every crossing. In such a ruling, ruling disks are either (globally) nested or disjoint, and hence such a ruling must be normal. This type of ruling is equivalent to an "admissible 0-resolution" of a front, as studied by Ng in [16]. A 0resolution of a front is the diagram obtained by "smoothing" crossings as in Figure 3. The aforementioned paper shows that alternating links have Legendrian representatives with normal -but not necessarily orientedrulings that switch at each crossing. In Section 3, below, we will revisit a result of Tanaka which implies that positive links also admit oriented normal rulings of this form.

Lagrangian Handle Attachment.
In this subsection, we describe constructions for Lagrangian fillings of Legendrian links in the standard contact R 3 . A Lagrangian submanifold L of the symplectization is a Lagrangian cobordism between the Legendrian submanifolds Λ ± ⊂ R 3 if there exists a pair of real numbers T − < T + such that Figure 4. Diagram moves corresponding to attaching a 0handle and an oriented 1-handle.
As defined in the introduction, a Lagrangian filling of a Legendrian link is a cobordism from the empty set to the given link. Note that stacking one Lagrangian cobordism on top of another results in a new Lagrangian cobordism.
The following tool allows us to construct Lagrangian fillings with a prescribed Legendrian link as their boundary. It has been adapted from Theorem 4.2 of [4], though this result appears in slightly different forms in the work of Ekholm-Honda-Kálḿan [9] and of Rizell [17]. We will call a filling decomposable if it can be split into cobordisms arising as in Theorem 2.2. Hereafter, Theorem 2.2 will be our primary means for producing links with Lagrangian fillings.

Positive Knots are Fillable
In this section, we prove Theorem 1.1. As mentioned above, the key tool in the proof is the existence of an oriented ruling in which all crossings are switched. The following lemma, essentially due to Tanaka [25], shows that we may assume the existence of such a ruling for a positive link. Proof. In [25], Tanaka extends an earlier result of Yokota [26] to show that every positive link has a Legendrian representative Λ with the following properties: (1) Every crossing has both strands oriented to the left (in particular, every crossing is positive); and (2) The 0-resolution of Λ consists of disjoint (but possibly nested) closed curves, each of which contains exactly one left cusp and one right cusp. By (1), the smoothing of each crossing in the 0-resolution respects the orientation of Λ. Thus, the strands which connect left and right cusps in the 0-resolution correspond to ruling paths in an oriented normal ruling that switches at each crossing.
We now prove that all positive links are fillable.
Proof of Theorem 1.1. By Lemma 3.1, it suffices to consider a fixed front Λ + that admits an oriented normal ruling in which all crossings are switched. The proof proceeds by induction on the number of crossings. If Λ + has no crossings, then it must be a disjoint union of maximal Thurston-Bennequin unknots. Such a link is fillable by the 0-handle construction in Theorem 2.2.
For the inductive step, suppose that every Legendrian whose front diagram has fewer crossings than Λ + and that admits an oriented normal ruling in which all crossings are switched is fillable. We begin by showing that the ruling of Λ + must have a switch with a neighborhood equivalent to one of the topmost diagrams in Figure 5 (up to an overall reversal of orientation or reflection through the horizontal). First, each switch occurs at a positive crossing, so the crossing paths must have the same horizontal direction. Second, companion paths are oriented in the opposite direction to the crossing paths, so the two companion strands must also have the same horizontal direction. Finally, because the ruling switches at each crossing, the ruling disks are either nested or disjoint. Any switch along the boundary of an innermost ruling disk will have the desired form.
Next, we use the crossing found above to construct a Legendrian link Λ − with the following properties: (1) The fronts of Λ + and Λ − are identical outside of a neighborhood of the crossing; (2) Near the crossing, the front of Λ − is of one of the forms depicted at the bottom of Figure 5; (3) Λ − has an oriented normal ruling with all crossings switched; and (4) There exists a Lagrangian cobordism from Λ − to Λ + . Only the last condition needs some verification, which is carried out in Figure 5.
Clearly, the Legendrian Λ − satisfies the inductive hypothesis and has one less crossing than Λ + . It follows that Λ − is fillable, and hence, by condition (4), that Λ + is fillable as well. The theorem follows.
Remark 3.2. We say that an oriented ruling of a front has an unlinked resolution if the result of performing 0-resolutions (see Figure 3) at all switches of the ruling results in an unlink, all of whose components are maximal tb unknots. Using this language, we see that the proof above actually gives a stronger result than Theorem 1.1: a smooth knot type is fillable if it has a Legendrian representative whose front has an oriented ruling with an unlinked resolution.

(Strong) Quasi-positivity and Fillability
We end this paper by providing the examples necessary to prove Theorem 1.2. The first example proves the first part of the theorem, namely that there exists a fillable quasi-positive knot that is not strongly quasi-positive. Example 4.1. As shown in Figure 6, the mirror of the 9 46 knot is fillable, and hence quasi-positive. On the other hand, since this knot is a non-trivial The example for the second part of the theorem -that there exists a non-fillable strongly quasi-positive knot -is somewhat more complicated. It was originally brought to light by Stoimenow [24,Example 3] in the context of disproving a conjecture of Fiedler on an upper bound for the minimal degree of the Jones polynomial of a (quasi-positive) knot in terms of a band representation.
On the other hand, we compute 3 that the degree in the framing variable of the HOMFLY polynomial of K is −10. Thus, the HOMFLY bound on the Thurston-Bennequin number would be tb(K) ≤ 9. Clearly, this bound is not sharp. As argued after Conjecture 1.3, however, if K were fillable, then the HOMFLY bound would be sharp. We must conclude, then, that K is not fillable.