Invariants for Legendrian knots in lens spaces Joan E. Licata Max-Planck-Institute fu¨r Mathematik/ Stanford University 9 0 [email protected] 0 2 January 27, 2009 n a J 7 Abstract 2 InthispaperwedefineinvariantsforprimitiveLegendrianknotsinlensspacesL(p,q),q 6=1. ] The main invariant is a differential graded algebra (A,∂) which is computed from a labeled N Lagrangian projection of the pair (L(p,q),K). This invariant is formally similar to a DGA G definedbySabloffwhichisaninvariantforLegendrianknotsinsmoothS1-bundlesoverRiemann h. surfaces. The second invariantdefined for K ⊂L(p,q) takes the form of a DGA enhanced with t a free cyclic group action and can be computed from the p-fold cover of the pair (L(p,q),K). a m [ 1 Introduction 1 v Endowing a three-manifold with a contact structure refines the associated knot theory by intro- 6 2 ducing new notions of equivalence among knots, and these in turn require invariants sensitive to 2 the added geometry. In addition to more classical numerical invariants, invariants taking the form 4 of differential graded algebras (DGAs) have seen success in distinguishing Legendrian non-isotopic . 1 knots in a variety of contact manifolds. 0 9 The first DGA invariants were developed for Legendrian knots in the standard contact R3. 0 Chekanov constructed a combinatorial invariant, and an equivalent invariant was introduced inde- : v pendently in a geometric context by Eliahsberg [Che02], [Eli98]. In the former case, the algebra i X is generated by the crossings in a Lagrangian projection, and the boundary map counts immersed r discs in the diagram. In Eliashberg’s relative contact homology, the algebra is generated by Reeb a chords and the differential counts rigid J-holomorphic curves in the symplectization of R3. The two constructions were shown to produce the same DGA in [ENS02]. In [Sab03], Sabloff adapted further work of Eliashberg, Givental, and Hofer to construct a combinatorial DGA for Legendrian knots in a class of contact manifolds characterized by their distinctive Reeb dynamics [EGH00], [Sab03]. His algebra is again generated by Reeb chords, but he introduces additional technical machinery in order to handle periodic Reeb orbits. Sabloff’s invariant is defined for smooth S1 bundles over Riemann surfaces, a class of manifolds which includes S3 and L(p,1), but does not admit other lens spaces. In this paper we develop an invariant for Legendrian knots in the lens spaces L(p,q) for q 6= 1 with the unique universally tight contact structure. For primitive K ⊂ L(p,q), we define a labeled diagram to be the Lagrangian projection of the pair (L(p,q),K) to (S2,Γ), together with some ancillary decoration which uniquely identifies the Legendrian knot. Numbering the crossings of Γ 1 from one to n, we consider the tensor algebra on 2n generators: A = T(a ,b ,...a ,b ). 1 1 n n We equip this algebra with a differential ∂ : A → A counting certain immersed discs in (S2,Γ). The algebra is graded by a cyclic group, and the boundary map is graded with degree −1. The pair (A,∂) is a semi-free DGA, and the natural equivalence on such pairs is that of stable tame isomorphism type. Our main theorem is the following: Theorem 2. Up to equivalence, the semi-free DGA (A,∂) is an invariant of the Legendrian type of K ⊂ L(p,q). The proof of Theorem 2 applies Sabloff’s invariant to a freely periodic knot K ⊂ S3 which is a p-to-one cover of K ⊂ L(p,q). The Legendrian type of K is an invariant of the Legendrian type of K, so Sabloff’s invariant for K is therefore also an invariant of K (Propositieon 1). In order to prove Theorem 2, we endow Sabloff’s DGA with addietional structure related to the covering transformations. e Given K in L(p,q) with q 6= 1, let (A,∂) denote Sabloff’s low-energy DGA for the knot K in S3. The algebra (A,∂) may be enhanced with a cyclic group action γ : Z × A → A which p commutes with the boundary map. We deefinee a notion of equivariant equivalence on DGAs witeh such actions in Sectione5e, and we associate to K the equivariant DGA (A,γ,∂). Oureseconed main theorem asserts that this is also an invariant of the Legendrian knot in the lens space. e e Theorem 3. The equivalence class of the equivariant DGA (A,γ,∂) is an invariant of the Legen- drian type of K. e e The major technical work of the paper lies in proving Theorem 3, and this occupies Section 5. The proof of Theorem 2 identifies (A,∂) with a distinguished Z -equivariant subalgebra of (A,∂) p andfollows asaconsequenceofTheorem3. Thefinalsection contains examples computedforknots in L(3,2) and L(5,2). e e Finally, we note that although the arguments in this paper are developed for primitive knots in L(p,q), they infactconstructinvariants forany Legendrianknotinalensspacewhichis covered by a Legendrian knot in some L(p,1). In this adaptation, L(p,1) replaces S3 as the contact manifold where Sabloff’s invariant is defined. I would like to thank JoshSabloff for helpfulcorrespondencein thecourse of writingthis paper. A portion of this work was conducted while visiting the Max Planck Institute for Mathematics in Bonn, Germany, and their hospitality and support are much appreciated. 2 Background Thissection contains abriefsummaryof thebasicdefinitionsfromcontact geometry andtheir real- izations inthreeexamples: thestandardcontact R3,S3,andL(p,q). Amorethoroughintroduction to the topic is provided in [Etn03] or [Gei08]. 2.1 Basic definitions A contact structure ξ on a three-manifold M is an everywhere non-integrable 2-plane field. A non-degenerate one-form α defines a contact structure by ξ = kerα at each point of M. Two α 2 contact manifolds (M ,ξ ) and (M ,ξ ) are contactomorphic if there is a diffeomorphism between 1 1 2 2 the manifolds which takes contact planes to contact planes. Definition 1. Given a contact form α, the Reeb vector field is the unique vector field X which satisfies α(X) = 1 dα(X,·) = 0. Integral curves of X are known as Reeb orbits, and they inherit an orientation from X. Definition 2. A knot K in (M,ξ) is Legendrian if its tangent lies in the contact plane at each point. Two Legendrian knots are equivalent if they are isotopic through Legendrian knots. In general, two knots which are topologically equivalent may not be Legendrian equivalent; any topological isotopy class of knots will be represented by countably many Legendrian isotopy classes. Definition 3. The Lagrangian projection of a contact manifold (M,ξ ) is the quotient space of M α whichcollapses each Reeborbitofαtoapoint. IfK isaLegendrianknotinacontact manifold,the Lagrangian projection of K is the image of the knot under Lagrangian projection of the manifold. If K is a Legendrian knot in (M,ξ), a Reeb chord is a segment of a Reeb orbit with both endpoints on K. In the Lagrangian projection, a Reeb chord with distinct endpoints will map to a crossing in the knot projection. 2.2 First example: R3 The standard contact structure ξ on R3 is induced by the contact form std α = dz−ydx. std The Reeb vector field on (R3,ξ ) has trivial dx and dy coordinates at every point, so the Reeb std orbits are vertical lines. Thus, the Lagrangian projection is simply projection to the xy-plane. 2.3 Second example: S3 S3 sits inside R4 as the unit sphere: S3 = {(r ,θ ,r ,θ )|r2+r2 = 1}. 1 1 2 2 1 2 The torus r = 1 = r separates S3 into two solid tori, and it will be convenient to treat this 1 √2 2 torus as a Heegaard surface. The curves r = 0 and r = 0 are the core curves of the Heegaard 1 2 tori, and the complement of the cores is foliated by tori of fixed r . i The standard tight contact structure on S3 is 1 α = (r2dθ +r2dθ ). 0 2 1 1 2 2 The punctured manifold (S3−{p},ξ ) is contactomorphic to (R3,ξ ), but the Reeb dynamics are 0 std quite different. In particular, the Reeb orbits of α are (1,1) curves on each torus of fixed r . This 0 i foliation of S3 by circles gives the Hopf fibration of S3, and Lagrangian projection in (S3,ξ ) is 0 projection to the S2 base space of this fibration. Note that the core curves are each Reeb orbits, and their images under Lagrangian projection are the poles of the two-sphere. The contact form α also induces a curvature form Ω on the S2 base space; for the standard contact structure, this is just the Euler class of the bundle, where S3 is viewed as the unit sphere in R4. [Gei08]. 3 2.4 Third example: Lens spaces Define F : S3 → S3 by p,q 2π 2qπ F (r ,θ ,r ,θ )= (r ,θ + ,r ,θ + ). (1) p,q 1 1 2 2 1 1 2 2 p p The map F generates a cyclic group of order p, and the quotient of S3 by the action of p,q this group is the lens space L(p,q). Thus π : S3 → L(p,q) is a p-to-one covering map. Since F preserves the contact structure on S3, π induces a contact structure on L(p,q) [BG]. The p,q Reeb orbits of (L(p,q),ξ ) again foliate the manifold by circles, and the Lagrangian projection of p,q (L(p,q),ξ ) is a two-sphere. As an S1 bundle over S2, L(p,q) is smooth if and only if q = 1. p,q Definition 4. A knot K in S3 is freely periodic if it is preserved by a free periodic automorphism of S3. e The map in Equation 1 has order p, so if K is freely periodic with respect to F , then π(K) p,q is a knot in L(p,q). Conversely, any K in L(p,q) which is primitive in H (L(p,q)) has a freely 1 periodic lift K ⊂ S3. (Knots which are not primeitive will lift to links in S3.) This definition makees sense in both the topological and contact categories; with respect to the contact structures defined above, K is Leegendrian if and only if K is Legendrian. An explicit construction of a freely periodic lift is describedin Section 6.2 of [GRS08], and we refer the reader to [HLN06] or [Ras07] for a fuller treatment of freely periodic knots. e Throughout the paper, each topological manifold will be equipped with the contact structure associated to itin this section; wewill writeonlyS3 andL(p,q) forthe contact manifolds (S3,ξ ), std and (L(p,q),ξ ). Furthermore, tildes will be used to distinguish objects in S3 from their coun- p,q terparts in L(p,q); thus Γ will denote the Lagrangian projection of a knot K in S3, whereas the Lagrangian projection of K ⊂ L(p,q) will be denoted by Γ. e e 3 Differential graded algebra invariants for Legendrian knots In this section we introduce Sabloff’s DGA invariant for Legendrian knots in smooth S1 bundles over Riemann surfaces. We begin by defining differential graded algebras and the relevant notion of equivalence among them. 3.1 Equivalence of semi-free DGAs Let V = SpanZ {x1,x2,...xn}. Define 2 ∞ A = T(x ,x ,...x ) = V n 1 2 n ⊗ n=0 M to be the tensor algebra on the elements {x ,x ,...x }. If V is graded by a cyclic group G so that 1 2 n the x are homogeneous, this induces a cyclic grading on A via the rule |x x |= |x |+|x |. When i i j i j ∂ : A → A is a degree −1 map satisfying ∂2 = 0 and the Leibnitz rule ∂(ab) = (∂a)b + a(∂b), then the pair (A,∂) is a semi-free differential graded algebra (DGA). The modifier “semi-free” emphasizes that we keep track of the preferred generators {x }n , which will be important in i i=1 defining DGA equivalence. 4 An elementary automorphism of A is a map gi : A → A such that x +v , for v ∈T(a ,....xˆ ...b ) if j = i gi(x ) = i i i 1 i n j (xj if j 6= i. When v is homogeneous in the same grading as x , we say that gi is a graded elementary automor- i i phism. A graded tame automorphism is a composition of graded elementary automorphisms. Given a DGA (A,∂) = (T(x ,...x ),∂), let E = T(e ,e ) be a DGA which is graded by the 1 n 1 2 same cyclic group and satisfies ∂ e = e and ∂ e = 0. A stabilization of A is the differential 1 2 2 E E graded algebra (T(x ,...x ,e ,e ),∂ ∂ ). 1 n 1 2 E Definition 5. Two semi-free differe`ntial graded algebras (A ,∂ ) and (A ,∂ ) are equivalent if 1 1 2 2 some stabilization of (A ,∂ ) is graded tame isomorphic to some stabilization of (A ,∂ ). 1 1 2 2 We will have reason to consider DGAs equipped with an action of a cyclic group Z , so we p extend the notion of equivalence to one respecting the group action. The cyclic group Z should p not be confused with the cyclic group G which grades the algebra. Definition 6. An equivariant DGA (A,γ,∂) is a semi-free DGA (A,∂) together with an automor- phism γ :A → A of order p such that ∂ ◦γ = γ◦∂ and |γx| = |x|. Definition 7. Suppose that (A,γ,∂) is an equivariant DGA, where A = T(x ,...x ). A free Z 1 n p stabilization of (A,γ,∂) is the equivariant DGA (A Ep,γ,∂ ∂ p), where E • A Ep = T(x ,...x ,e ,e ,...e ,e ,...e `); ` 1 n 1,1 1,2 1,p 2,1 2,p • γ(e` ) = e ; j,i j,i+1 • ∂ pe1,i = e2,i; E • ∂ pe2,i = 0. E Definition 8. AZ elementary isomorphism is aγ-equivariant mapf : (A,γ,∂) → (A,γ,∂) which p can be written as f = g1◦g2◦...◦gp, where each gi is an elementary isomorphism. If f is graded, we say it is a graded Z elementary isomorphism. A composition of Z elementary isomorphisms p p is a Z tame isomorphism. p Definition 9. Two equivariant DGAs (A ,γ ,∂ ) and (A ,γ ,∂ ) are Z equivalent if they have 1 1 1 2 2 2 p free Z stabilizations which are graded Z tamely isomorphic. p p 3.2 Sabloff’s DGA for knots in S1 bundles over Riemann surfaces In [Sab03], Sabloff considers contact manifolds whose Reeb orbits are the fibers of a smooth S1 bundle over a Riemann surface. For a Legendrian knot K in such a manifold, he defines an algebra generatedbytheReebchordswithbothendpointsonK. SinceeachReeborbitisperiodic,thereare infinitely many such chords, and he also defines a finiteely-generated low-energy algebra generated by chords which are strictly shorter than the fiber.eThe low-energy algebra sits inside the full invariant as a subalgebra, but the equivalence type of the low-energy algebra is also an invariant of K. The following section introduces the the low-energy algebra (A,K) for Legendrian knots in S3, and we refer the reader to [Sab03] for a description of the full invariant. e e e 5 3.2.1 Labeled Lagrangian diagram Let K ⊂ S3 bea Legendrian knot, and denote the Lagrangian projection of K to S2 by Γ. Number the crossings of the diagram from 1 to n, and associate two generators a and b to the ith crossing. i i Thesee correspond to the complementary short chords in the fiber which ientersects thee crossing strands of K. Because the Reeb orbit is oriented, each chord identifies the crossing strands locally as “sink”and “source”. Select a preferredchord and indicate this choice with a plussign in thetwo (opposite) qeuadrants where traveling sink-to-source orients the quadrant positively. Furthermore, assign each quadrant either a+ and b or a and b+ as indicated in Figure 1. Note that signs i −i −i i are used in two distinct ways; a “positive quadrant” will always mean one marked with a “+” to denote the preferred chord, and each generator x labels every quadrant of the associated crossing as either x+ or x . − a b + b _ + a _ a + b _ b + b + a + a _ Figure 1: Labels at a crossing on Γ. The “+” in the right-hand diagram indicates that a is the preferred chord. Definition 10. If Γ is a labeled diagram with n crossings, define A(Γ) to be the tensor algebra generated by the associated Reeb chords: e e e A(Γ) = T(a ,b ,...a ,b ). 1 1 n n The following definitions will proveeeuseful in defining the defect and the boundary map: Definition 11. If x is a generator of A(Γ), let l(x ) denote the length of the associated chord in i i S3, where the length of an S1 fiber is normalized to 1. e e We extend this to a length function l on words written in the signed generators a and b . ′ ±i ±i Let ǫ(a+) = ǫ(b+) = 1 and ǫ(a )= ǫ(b ) = −1. If w is a word in the signed generators x , define i i −i −i ±i l (w) = ǫ(x )l(x ). ′ ±i i ± xXi ∈w Definition 12. Let (Σ,∂Σ) be a disc with m marked points on the boundary. An admissible disc is a map f :(Σ,∂Σ) → (S2,Γ) which satisfies the following: 1. each marked point maps to a crossing of Γ; e 2. f is an immersion on the interior of Σ; e 3. f extends smoothly to ∂Σ away from the marked points; 6 4. f(∂Σ) has a corner at each marked point, and f(Σ) fills one quadrant there. Two admissible discs f and g are equivalent if there is a smooth automorphism φ: Σ→ Σ such that f =g◦φ. Let R be a component of S2−Γ. To each corner of R, one may associate the signed generator corresponding to the preferred chord of the crossing, where the sign is dictated by the quadrant filled byR. Traveling counterclockweise around∂R andreadingoff theselabels definesacyclic word w(R). (See Figure 2 for an example.) + + a - f b + 1 1 b - b - 3+ + 2a + a + + 2 3 + Figure 2: An admissible disc with w(R) = a b a+. This disc could represent three different −1 −2 3 boundary words: w(f,b )= b b ; w(f,a )= b a ; or w(f,a ) = a b . 1 2 3 2 3 1 3 1 2 Definition 13. Let f be an admissible disc whose image is R. The defect of R is given by: 1 n(f)= f Ω+l (w(R)). (2) ∗ ′ 2π ZΣ Geometrically, the defect encodes the interaction between the knot and the fiber structure. Without this decoration, Γ does not specify even the topological type of the knot, as displacement in the Reeb direction is obscured by the projection. The curve ∂R lifts to S3 as a simple closed curvecomposedofalternaeting LegendrianandReebsegments, andthedefectmeasuresthewinding number of this lifted curve around the fiber with respect to an appropriate trivialization. Together with the signs at each crossing, the defects of components of S2−Γ determine the Legendrian type of the knot. e -1 1 + 0 + + 0 -1 -1 + Figure 3: Two labeled diagrams for the same Legendrian unknot in S3. The defect extends additively to unions of regions counted with multiplicity, so Equation 2 holds for any admissible disc f. Since S2 is simply connected, one may also define the defect of the knot n(K) to be the defect of any contracting disc bounded by the projection of K. 3.2.2 Ge radings e A capping path for a generator x is a path along K in Γ which begins and ends adjacent to the i same x+ quadrant. For each crossing, one of a or b will have two capping paths, and the other i i i will have none. The rotation number of a cappingepathefor x is the number of counterclockwise i 7 rotations performed by the tangent vector, computed as a winding number in a trivialization over a contracting disc in S2. Taking the edges at a crossing to be orthogonal, this value lies in Z− 1, 4 and we denote it by r(x ). i Suppose that f :Σ → S2 is an admissible disc such that f(∂Σ) is a capping path for x . Then i the grading of x is given by the following: i 1 |x | = 2r(x )− +4n(f). (3) i i 2 If y is the other generator at the same crossing, i |y | = 3−|x |. (4) i i These gradings are well-defined modulo 2r(x )+4n(K). i 3.2.3 The algebra (A(Γ),∂) e Definition 14. Let f : Σ → S2 be an admissible disc with one corner filling a quadrant labeled e e e x+. The boundary word w(f,x ) is the concatenation of the y generators associated to the other i i j− quadrants filled by f(Σ), read counterclockwise around ∂Σ. w(f,x )= y y ...y . i 2 3 m See Figure 2 for an example. Definition 15. If f :Σ → S2 is an admissible disc, the x defect n˜ (f) is given by i xi m 1 n˜ (f) = f Ω+l(x )− l(y ). xi 2π ∗ i j ZΣ j=2 X Note that the x defect of an admissible disc may differ from the defect of its image in the i diagram, as the two are computed by associating (possibly) different words to the same disc. To compute n˜ (f), add one to n(f) if x+ occupies a non-positive quadrant, and subtract one from xi i n(f) for each y in w(f,x ) which occupies a positive quadrant. Thus both types of defects may j− i be computed from the labeled diagram without further data regarding the lengths of chords. Definition 16. The differential ∂ :A(Γ)→ A(Γ) is defined by on the generator x by i e e∂xei = e e w(f,xi), f:n˜xXi(f)=0 e and ∂ extends to other elements in A(Γ) via the Leibnitz rule ∂(ab) = (∂a)b+a(∂b). Theorem 1 ([Sab03] Proposition 3.8, Theorem 3.11, Corollary 3.16). The boundary map in Defi- e e e e e e nition 16 satisfies ∂2 = 0, and the stable tame isomorphism type of (A(Γ),∂) is an invariant of the Legendrian knot type of K in S3. e e e e e 8 4 Invariants for Legendrian knots in lens spaces As noted above, Sabloff’s invariant is definedfor contact manifolds which are smooth S1 bundles,a class whichexcludesthelensspacesL(p,q)forq 6= 1. Althoughtheydonotinducesmoothbundles, the Reeb orbits of these lens spaces nevertheless define an S1 bundle structure, and this similarity is strong enough to permit a DGA invariant (A,∂) computable from the Lagrangian projection to S2. The invariant is formally similar to Sabloff’s invariant (A,Γ) for knots in S3, and in fact, the proof of invariance exploits the covering relationship between these manifolds. Except if otherwise indicated, in the remainder of the papeereevery lens space L(p,q) is assumed to have q 6= 1. 4.1 The DGA (A,∂) Let K be a knot in L(p,q) which generates H (L(p,q)). Following Rasmussen, we call such knots 1 primitive [Ras07]. If K is a primitive Legendrian knot, we begin by defining a labeled Lagrangian diagram. At the ith crossing of Γ, mark each quadrant with a+ and b or with a and b+ as in i −i −i i Figure 1. At each crossing, indicate a preferred choice of chord by decorating a pair of opposite quadrants with plus signs. Recall that to K, we may associate its freely perioidic lift K ⊂ S3. The p-fold covering map π : (S3,K) → (L(p,q),K) descends to a p-to-one branched cover of Lagrangian projections π :(S2,Γ) → (S2,Γ), where the branch points are the images of thee core curves r = 0 for i= 1,2. i ∗ Thus a choiceeof preferred chords in Γ lifts to a choice of preferred chords in Γ. Let R be a region in (S2−eΓ), and let f : Σ → S2 be an admissible disc whose image is π 1(R). Define the defect of − R to be n(R)= 1n(f). ∗ e p A labeled diagram for K is a generic Lagrangian projection Γ decorated with preferred chords and defects which are compatible with a labeled diagram for K as described above. Definition 17. LetΓbealabeleddiagramforaLegendrianknotK ⊂ L(p,q). IfΓhasncrossings, e define A(Γ) = T(a ,b ,...a ,b ). 1 1 n n Ifx is agenerator of A(Γ), choose aliftx ∈ π 1(x )and definethegradingofx by|x | =|x |. i ∗i − i i i ∗i ∗ Thisvalueis independentof thechoice oflift, andA(Γ)is gradedby thesamecyclic groupas A(Γ). Remark The grading can also be defined intrinsically. Given a labeled diagram, consider a capping path for x which has winding number p with respect to the poles. With only seligeht i modification, the formulae in Equations 3 and 4 can be used to compute the grading directly from Γ. Definitions 12 and 14 may be applied verbatim in the context of labeled diagrams for knots in L(p,q). Definition 18. The differential ∂ :A(Γ)→ A(Γ) is defined on generators by ∂x = w(f,x ), i i f:n˜xXi(f)=0 wherethesumisoveradmissiblediscswhichsatisfytheadditionalconditionthatf(∂Σ)haswinding number p with respect to the poles of S2. Extend ∂ to other elements in A(Γ) via the Leibnitz rule. 9 Theorem 2. Up to equivalence as a semi-free DGA, (A(Γ),∂) is an invariant of the Legendrian knot type of K ⊂ L(p,q). In order to prove Theorem 2, we will study the relationship between (A(Γ),∂) and (A(Γ),∂). It is clear that the Legendrian type of the freely periodic lift K ⊂ S3 is an invariant of the Legendrian type of K, so Sabloff’s construction has the following easy consequence: e e e e Proposition 1. The stable tame isomorphism type of (A(Γ),∂) is an invariant of the Legendrian isotopy class of K. e e e However, a stronger notion of equivalence yields a more interesting invariant, and the next section shows that we may associate an equivariant DGA to the freely-periodic lift of K. 4.2 The Z action on (A(Γ),∂) p Theorem 3.15 of [Sab03] states that the equivalence type of (A(Γ),∂) is independent of the choice of preferred chords, but we will restrict attention to diagrams where the signs at each crossing are preserved by 2π rotation of (S2,Γ). e e e p Lemma 1. If K is a Legendrianeknot in L(p,q), then there is a natural automorphism γ :A(Γ) → A(Γ) with order p such that ∂ ◦γ = γ◦∂, and |γx| = |x|. e e Proof. Fix a representative of the isotopy class of K and lift this to the freely periodic knot K. e e e e The Lagrangian projection of (S3,K) is invariant under 2π rotation about the axis through the p e points representing the fibers r = 0 for i = 1,2. If the projection of K is not generic, any i e local perturbation of K will lift to p local perturbations of K, maintaining the contact covering relationship between (S3,K) and (L(p,q),K) while removing singularitiesein the projection. In particular, K (or equivalently, K) may be assumed disjoint freom the cores of the Heegaard tori. If the Reeb chord x is ae generator of A, then π 1(x) is a free Z orbit of generators of A(Γ). − p This relationship descends to tehe Lagrangian diagrams, via the p-fold branched covering map π : (S2,Γ) → (S2,Γ). Since capping paths for crossings in a single orbit are permuted byetehe ∗ cyclic action, each member of the orbit has the same grading. Similarly, any disc which represents a term inethe boundary is part of an orbit of p discs. This proves that the Z action on A(Γ) p commutes with the differential. e e Remark Recall that Sabloff’s invariant is defined for knots in lens spaces L(p,1). The above discussion highlights another sense in which this case is exceptional. When q = 1, the map π : ∗ (S2,Γ) → (S2,Γ) induced on Lagrangian projections is one-to-one. In this case, for any point x on Γ, the preimage F 1(x) consists of p points on K. p−,1 e Theorem 3. If K is a Legendrian knot in L(p,q), then the equivariant DGA (A(Γ),γ,∂) is an e e invariant of K, up to Z equivalence. p e e e The proof of Theorem 3 appears later, but we note a corollary of the statement here: Corollary 1. Let Aγ(K) be the subalgebra of (A(Γ,γ,∂) fixed by the Z action: p Aγ(K) = {a ∈ A(Γ)|γa = a}. e e e The subalgebra Aγ(K) isa subcomplex and the homology of Aγ(K)isan invariant of the Legendrian e e type of K. 10