Augmentations, annuli, and Alexander polynomials
AAUGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS
LU´IS DIOGO AND TOBIAS EKHOLM
Abstract.
The augmentation variety of a knot is the locus, in the 3-dimensional coefficient spaceof the knot contact homology dg-algebra, where the algebra admits a unital chain map to thecomplex numbers. We explain how to express the Alexander polynomial of a knot in terms of theaugmentation variety: it is the exponential of the integral of a ratio of two partial derivatives. Theexpression is derived from a description of the Alexander polynomial as a count of Floer strips andholomorphic annuli, in the cotangent bundle of Euclidean 3-space, stretching between a Lagrangianwith the topology of the knot complement and the zero-section, and from a description of theboundary of the moduli space of such annuli with one positive puncture.
Contents
1. Introduction 2Acknowledgements 42. The Alexander polynomial and gradient flows 42.1. Dynamical definition of the Alexander polynomial 42.2. The Alexander polynomial via Morse theory 53. Flow loops, flow lines, and holomorphic curves 83.1. Models of L K and M K Date : May 21, 2020.2010
Mathematics Subject Classification. a r X i v : . [ m a t h . S G ] M a y LU´IS DIOGO AND TOBIAS EKHOLM Introduction
Let K Ă R be a knot and let ∆ K p µ q denote its Alexander polynomial. We study the relationbetween ∆ K p µ q and holomorphic curve invariants of Lagrangian submanifolds of T ˚ R naturallyassociated to K . We first introduce the holomorphic curve invariants and then state our mainresult and discuss its ramifications.Let ST ˚ R denote the unit co-sphere bundle of R . The restriction of the Liouville form p dq to ST ˚ R is a contact form and the submanifold of unit covectors over K that annihilates its tangentvector is a Legendrian torus Λ K Ă ST ˚ R .We describe two Lagrangian fillings L K and M K in T ˚ R naturally associated to K . The first, L K , is the Lagrangian conormal of K that consists of all (not necessarily unit length) co-vectorsalong K that annihilate its tangent vector. The conormal L K is diffeomorphic to a solid torus, L K « S ˆ R . The second, M K , is obtained from L K and the zero-section R : L K intersects R cleanly along K and the Lagrangian M K is obtained by Lagrange surgery on this clean intersection.The Lagrangian M K is diffeomorphic to the knot complement M K « R z K .The Chekanov–Eliashberg algebra A p Λ K q of Λ K is called the knot contact homology of K . It isa dg-algebra freely generated by the Reeb chords of Λ K with coefficients in C r H p ST ˚ R , Λ K qs , thegroup algebra of the second relative homology of Λ K . Pick a homology basis p t, x, p q , where t is agenerator of H p ST ˚ R q , x maps to the longitude of the knot under the connecting homomorphism,and p to the meridian. Here we use the natural identification of Λ K with the boundary of atubular neighborhood of K . The differential in the dg-algebra counts rigid holomorphic disks inthe symplectization R ˆ ST ˚ R with boundary in R ˆ Λ K , with one positive and several negativepunctures at Reeb chords.An augmentation of A p Λ K q is a dg-algebra map (cid:15) : A p Λ K q Ñ C . Given such a chain map, thedga-differential induces a differential on the chain complex ker p (cid:15) q{ ker p (cid:15) q the homology of whichis called the (cid:15) -linearized homology of A p Λ K q . Denoting the collection of Reeb chords of degree 0by a , the full augmentation variety ˜ V K is the set of values p e t , e x , e p , a q P p C ˚ q ˆ C | a | for such anaugmentation. Let π : p C ˚ q ˆ C | a | Ñ p C ˚ q be the projection. The top-dimensional stratum V K of the Zariski-closure of π p ˜ V K q Ă p C ˚ q is the augmentation variety. The augmentation variety hascodimension at most one. It is well-known that the point p e x , e p , e t , a q “ p , , , q “ p , q lies in˜ V K for any K . We show the following result about the full augmentation variety in a neighborhoodof this point. Theorem 1.1.
For any knot K , the full augmentation variety ˜ V K is smooth and 2-dimensional ina neighborhood ˜ V K of p e x , e p , e t , a q “ p , q , with T p , q ˜ V K isomorphic to ker p dt q under d p , q π . Forevery augmentation in a neighborhood ˜ U K Ă ˜ V K of p , q , the corresponding linearized homologyhas rank one in degrees one and two and rank zero otherwise. Furthermore, the degree one and twohomologies form rank one holomorphic line bundles over ˜ U K . Theorem 1.1 is proved in Section 4.3. It implies that the augmentation variety V K Ă p C ˚ q contains a smooth 2-dimensional surface V K “ π p ˜ V K q through the point p , , q , parametrized bythe variables λ “ e x and µ “ e p . In particular, V K is of codimension at most one, and is cut outby a single polynomial, called the augmentation polynomial and denoted by Aug K p e x , e p , e t q .Since the knot complement Lagrangian filling M K induces augmentations of A p Λ K q , the line t x “ “ t u lies in V K . Pick a vector field y p e p q of chain-level representatives of generators of thedegree one linearized homology line bundle in Theorem 1.1, for points in tp , e p , qu X V K . Thepositive puncture of the disks we count lies at the chords in y p e p q and the counts are as follows.First, f x p e p q counts disks that contribute to the dg-algebra differential of y p e p q , augmented by M K at negative punctures, and with boundary passing through a fixed meridian curve. Second, f t p e p q counts disks that contribute to the dga-differential of y p e p q , augmented by M K at negative UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 3 punctures, and that pass through a dual of the homology class t at an interior point. More detailswill be given in Section 5. Our main result is the following. Theorem 1.2.
Let K Ă R be a knot, Aug K p e x , e p , e t q its augmentation polynomial, and ∆ K p e p q its Alexander polynomial. With notation as above, we have f x p e p q ‰ , except at finitely manypoints, and ∆ K p e p q “ p ´ e p q exp ˆż ´ f t p e p q f x p e p q dp ˙ (1.1) “ p ´ e p q exp ˆż ´ B t Aug K B x Aug K ˇˇˇˇ p x,t q“p , q dp ¸ , (1.2) where (1.2) holds provided B x Aug K | p x,t q“p , q ‰ . Theorem 1.2 is proved in Section 5. The proof uses several cobordisms of moduli spaces ofholomorphic curves. The starting point for the argument is Theorem 1.3 below. It interprets adynamical formula for the Alexander polynomial of a knot K Ă R , consisting of counts of gradientflow lines and loops in the knot complement, in terms of holomorphic annuli and strips in T ˚ R . See[Mil62, Fri83, HL99] for the dynamical formula, and Section 2.2 for an alternative Morse-theoreticproof. Let M δ K denote a shift of the exact Lagrangian knot complement along a generic closed1-form representing a generator of the first cohomology. The following result is proved in Section3.2 using flow tree techniques from [Ekh07]. Theorem 1.3.
Let K Ă R be a knot. For any almost complex structure on T ˚ R agreeing withthe standard almost complex structure along the zero-section R and for all sufficiently small shifts,the Alexander polynomial of K can be written as (1.3) ∆ K p e p q “ p ´ e p q ζ an p e p q ¨ τ str p e p q , where ζ an p e p q is the exponential of a generating function of holomorphic annuli in T ˚ R stretchingfrom R to M δ K and τ str p e p q counts holomorphic strips between R and M δ K . The right hand side of (1.2) in Theorem 1.2 makes sense also at points in the augmentationvariety V K , where p x, t q ‰ p , q . This gives a Q -deformation of the Alexander polynomial ( Q “ e t )that connects our work to topological string and physics invariants in low-dimensional topology.More precisely, this observation is the starting point for a geometric treatment and Q -deformationof the so called p Z -invariant of 3-manifolds, originating from a 3d-3d correspondence, see [GPV17,GPPV17], for knot complements as described in [EGG `
20, Section 4]. We describe this in Section7. For fibered knots, we prove in Proposition 7.9 that the quotient by 1 ´ e p of this Q -deformationof the Alexander polynomial, where Q “ e t , can be written as τ K “ exp ` B t U K p p, t q ˘ , where U K isa certain Gromov–Witten disk potential for the knot complement Lagrangian M K . In Conjecture7.10, we conjecture that a generalization holds true also for non-fibered knots. Remark . The logarithm of τ K can be interpreted as the dual coordinate s “ s p x, t q or s “ s p p, t q of the deformation coordinate t in an extended (holomorphic) Lagrangian augmentation variety in p C ˚ q . This extended augmentation variety is Lagrangian with respect to the symplectic form dx ^ dp ` dt ^ ds , and given by the generating function U K p p, t q .The paper is organized as follows. In Section 2, we recall a formula for the Alexander polynomialof a knot K Ă S in terms of gradient flow loops and gradient flow lines in the knot complement,and give a Morse-theoretic proof of that formula. In Section 3, we study the Lagrangians L K and M K and their non-exact deformations, analyze holomorphic disks and annuli with boundaries inthese Lagrangians and in R , and prove Theorem 1.3. In Section 4, we discuss coefficients in knotcontact homology, then study the augmentation variety of a knot, compute some linearized knot LU´IS DIOGO AND TOBIAS EKHOLM contact homology groups and prove Theorem 1.1. Section 5 is the main section of the paper. There,we study several cobordisms of moduli spaces of holomorphic curves and prove Theorem 1.2. Wealso relate the dependence of choices in that result with well-known properties of the Alexanderpolynomial. In Section 6, we present some examples to illustrate our results. In Section 7, wediscuss how Theorem 1.2 relates to physical invariants from the point of view of Gromov–Wittendisk potentials and Floer torsion. Section 7 is the only part of the paper that relies on abstractperturbations for holomorphic curves.
Acknowledgements.
LD thanks Paul Seidel for a suggestion that led to the start of this project.We thank Lenny Ng for helpful comments. LD was supported by the Knut and Alice WallenbergFoundation. TE was supported by the Knut and Alice Wallenberg Foundation and the SwedishResearch Council. 2.
The Alexander polynomial and gradient flows
In this section, we discuss how to express the Alexander polynomial in terms of counts of gradientflow lines and gradient flow loops of S -valued Morse functions. The results are well-known. Westate them in Section 2.1 and give a direct Morse-theoretic derivation in Section 2.2.2.1. Dynamical definition of the Alexander polynomial.
Let K Ă S be a knot and ν p K q atubular neighborhood of K . Let Y K “ S z K and ¯ Y K “ S z ν p K q . Then H p Y K q “ Z . Let p be agenerator of this group. Consider an S -valued Morse function f : ¯ Y K Ñ S such that f ˚ p p q “ r S s .If f has critical points, then we can assume that they all have index 1 or 2. Let g be a metric on¯ Y K . Let ∇ g f denote the g -gradient vector field of f and ϕ T its time- T flow. Assume that ∇ g f iseverywhere transverse to the boundary B ν p K q .By the Kupka–Smale theorem, see e.g. [MP82], for generic g , we can assume two further transver-sality conditions: stable and unstable manifolds of ∇ g f intersect transversely and every gradientflow loop γ of period T ą γ , the linearized Poincar´e return map p dϕ T q x , x P γ , does not haveany eigenvalue of modulus 1. This implies that γ has a neighborhood that does not contain anyother flow loops (apart from multiples of γ ).Let O denote the discrete set of flow loops of ∇ g f . Consider the zeta function(2.1) ζ loop p µ q “ exp ˜ ÿ γ P O σ p γ q m p γ q µ d p γ q ¸ P Q rr µ ss . Here, if γ has period T and x P γ , then σ p γ q P t˘ u is the sign of the determinant of 1 ´ p dϕ T q x , m p γ q is the largest integer such that γ factors through an m -fold cover S Ñ S , and r γ s “ d p γ q p on H p Y K q .The function f has an associated Novikov complex, that can be defined as follows. Consider themaximal Abelian cover of r Y K of ¯ Y K , with projection map π : r Y K Ñ ¯ Y K . Denote by µ the positivegenerator of the deck transformation group Z . The lift r f : r Y K Ñ R of f is a Morse function.Consider the free product over Q generated by the critical points of ˜ f graded by the Morse index.Elements of the free product can be thought of as infinite sums of critical points and we let C ˚ p ˜ f ; Q q be the subspace of half-infinite elements. More formally, ÿ x P Crit p f q ,a x P Q a x ¨ x P C ˚ p ˜ f ; Q q if, for any x such that a x ‰
0, t k ą | a µ ´ k ¨ x ‰ u ă 8 . UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 5
The action of Z on r Y K by deck transformations endows C ˚ p ˜ f ; Q q with a free action of the ringof formal Laurent series Q pp µ qq , consisting of sums with finitely many negative powers of µ andpossibly infinitely many positive powers. Here, µ is to be thought of as the exponential of thepositive generator of Z (or, equivalently, of the generator p P H p Y K q above). Since Q is a field, Q pp µ qq is also a field, isomorphic to the field of fractions of the formal power series ring Q rr µ ss . Thecomplex C ˚ p ˜ f ; Q q is a finite dimensional vector space over Q pp µ qq , generated by the critical pointsof f . Recall that we assume that all these critical points have index 1 or 2. The Morse torsion isdefined as(2.2) τ Morse p µ q : “ det M p µ q , where M p µ q is the matrix representing the differential À x P Crit p f q Q pp µ qqx x y d ÝÝÝÝÑ À y P Crit p f q Q pp µ qqx y y . If f has no critical points, then we say that τ Morse = 1.Results of [Mil62, Fri83, HL99] imply that the Alexander polynomial of K can be written as theproduct(2.3) ∆ K p µ q “ p ´ µ q ζ loop p µ q ¨ τ Morse p µ q . In Section 2.2, we give a Morse-theoretic derivation of this formula.
Remark . If K is a fibered knot, we can think of the Morse function f as coming from thefibration Σ Ñ S z K Ñ S , which endows the knot complement with an open book decompositionwith monodromy given by a map φ : Σ Ñ Σ. We can arrange for φ to be a symplectomorphismwith respect to an area form on Σ. In that case, O is the collection of generators of the fixed pointFloer homology of φ (see [Spa17]).For K fibered, the function ζ loop agrees with a Gromov–Taubes invariant of S ˆ S p K q , where S p K q is 0-surgery on K Ă S . More precisely, [Hut14, Lemma 2.11] and [Spa16] give explicitformulas for the contribution to this Gromov–Taubes invariant of a simple periodic orbit γ andits covers, depending on whether γ is elliptic or (positive or negative) hyperbolic. See Remark3.5 for a related point concerning holomorphic annuli. For a general knot K , see [Mar02] for anidentification of the product ζ loop ¨ τ Morse with a Seiberg–Witten invariant of S p K q .2.2. The Alexander polynomial via Morse theory.
In this section, we show that there existfunctions f and metrics g on ¯ Y K for which equation (2.3) holds. The formula then actually holdsfor generic p f, g q , since the right side is invariant under generic deformations, see [Hut02]. Althoughthe rest of the paper is independent of this section, we include it since it gives a direct relationbetween flow loops and holomorphic annuli and flow lines and holomorphic disks in Morse or Floertheory that seems to be of importance also more generally. We use the notation of Section 2.1.The idea is to break the flow loops and flow lines of ∇ g f in the previous section into sequencesof short flow lines, by suitably perturbing the function f : ¯ Y K Ñ S . Lifting the perturbed functionto ˜ Y K Ñ R and perturbing slightly, gives a Morse chain model for the homology of the Abeliancover r Y K , as a Q r µ ˘ s -module. (Before perturbing f , the lifted function could have arbitrarily longflow lines and we would not be able to use polynomial coefficients.) In this setting, the Alexanderpolynomial can be thought of as the degree 1 contribution to the Reidemeister torsion of thishomology.To be more specific, recall that r Y K is a maximal Abelian cover of ¯ Y K , which endows H k p r Y K ; Q q with the structure of a finitely generated Q r µ ˘ s -module, for every k . We will be interested inthe case k “
1. Since Q is a field, Q r µ ˘ s is a principal ideal domain. The module H p r Y K ; Q q istorsion and the Alexander polynomial ∆ K p µ q generates the corresponding order ideal in Q r µ ˘ s .This means that(2.4) H p r Y K ; Q q « Q r µ ˘ s{p P p µ qq ‘ . . . ‘ Q r µ ˘ s{p P m p µ qq and ∆ K p µ q “ P p µ q . . . P m p µ q , LU´IS DIOGO AND TOBIAS EKHOLM
F F ` F ´ ind 1ind 2 C p Y K q C p Y K q q C ˚ p F q p C ˚ p F q µ q C ˚ p F q Figure 1.
Schematic depiction of the graph of ˆ φ . The arrows indicate where thesummands of C ˚ p r Y K ; µ q are supported (when restricted to a fundamental domainfor the action of Z on r Y K by deck transformations).where P j p µ q are irreducible polynomials.Recall that f : Y K Ñ S is an S -valued Morse function which only has critical points of indices1 or 2, and that r f : r Y K Ñ R is its lift. Let θ be a regular value of f and let F “ f ´ p θ q . Inorder to create a µ -invariant Morse complex for r Y K , we consider a Bott–Morse perturbation of f ,denoted φ : Y K Ñ S , with a canceling pair of Bott maxima p F and minima q F on nearby preimages F ˘ “ f ´ p θ ˘ (cid:15) q , and we let ˜ φ : r Y K Ñ R denote its lift. After morsification we then get thefollowing Morse complex of ˆ φ : r Y K Ñ R : C ˚ p r Y K ; µ q “ q C ˚ p F q ‘ p C ˚ p F q ‘ C p Y K q ‘ C p Y K q , where all summands on the right are Q r µ ˘ s -modules. See Figure 1 for a schematic depiction. Wethink of F as a two dimensional handlebody with an associated Morse function. We can assumethat q C ˚ p F q is supported in degrees 0 and 1, and that the Morse differential on any index 1 criticalpoint s is B s “ m ´ m “ , where m denotes the unique index 0 critical point. The differential in this complex can be writtenas p C p F q ‘ C p Y K q B ÝÝÝÝÑ q C p F q ‘ p C p F q ‘ C p Y K q B ÝÝÝÝÑ q C p F q where B | q C p F q‘ C p Y K q ” B | p C p F q : p C p F q Ñ q C p F q is the map 1 ´ µ . Hence, H p r Y K ; µ q – Q r µ ˘ s{p ´ µ q – Q . Furthermore, the image of B is zero on the summad p C p F q , and H p r Y K , µ q isthe cokernel of the µ -module homomorphism(2.5) D “ B : p C p F q ‘ C p Y K q Ñ q C p F q ‘ C p Y K q . The proof that Morse homology is isomorphic to singular homology can be adapted to showingthat the homology H ˚ p r Y K ; µ q is isomorphic to H K p r Y K ; Q q as Q r µ ˘ s -modules. By (2.4), we can use H p r Y K ; µ q to compute the Alexander polynomial. Lemma 2.2.
Pick bases for the left and right hand sides of (2.5) , and let A p µ q be the correspondingmatrix representation of D . Then ∆ K p µ q “ det p A p µ qq . UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 7
Proof.
Since H p r Y K ; µ q – coker p D q , the lemma follows from a standard fact about finitely generatedtorsion modules over principal ideal domains. The idea is that ∆ K p µ q is the product of the invariantfactors of the Q r µ ˘ s -module H p r Y K ; µ q , and these invariant factors are the diagonal entries of theSmith normal form of the matrix of relations A p µ q . The product of these entries is the determinantof the matrix. (See e.g. [Lan02, Section III.7] for details.) (cid:3) Remark . Alternatively, one can use the following elementary argument to demonstrate Lemma2.2. If the right hand side of (2.5) has only one generator, the result is obvious. If there are n generators c j , j “ , . . . , n then the image of the differential gives relations ř nj “ a ij c j “
0. Usethe first equation to eliminate c , c “ ´ a p a c ` ¨ ¨ ¨ ` a n c n q . The remaining equations are for i, j ą ÿ j ą ˆ a ij ´ a i a a j ˙ c j “ . By induction, the ideal is then given by the determinant det p A q , which is equal the determinantof the modified version ¯ A obtained from A by Gauss eliminating the first column multiplying thefirst row by a . The resulting determinant is then linear in a and by cofactor expansiondet p A q “ det p M q ` ÿ i ě a p´ q i a i det p M i q , where M ij is the ij -minor of ¯ A . Consequently, the original generator is a det p A q “ ÿ i ě p´ q i a i det p M i q “ det p A q . Consider the Morse complex C ˚ p r Y K ; µ q , generated by critical points of ˆ φ and with differentialthat counts flow lines. Let d denote the count of flow lines from C p Y K q to C p Y K q . Let ψ F denotethe count from p C p F q to µ q C p F q , η the count from p C p F q to C p Y K q and ψ c the count from C p Y K q to µ q C p F q . The differential on the Morse complex is then represented as the µ -module map withmatrix D p µ q “ ˆ ´ µψ F ´ µψ c η d ˙ . Lemma 2.2 implies that(2.6) ∆ K p µ q “ det D p µ q . Note that, by setting µ “ K p µ q is det d . If K isfibered, then we can assume that f has no critical points, and recover the well-known fact that theAlexander polynomial is monic.In order to get a geometric interpretation of det D p µ q , we first express it as a product of twodeterminants. At this point, we use the inclusion Q r µ ˘ s Ă Q pp µ qq :det ˆ ´ µψ F ´ µψ c η d ˙ “ det ˆ ´ µψ F ´ µψ c µηψ F d ` µηψ c ˙ “ det ˆ ´ µψ F ´ µψ c µ ηψ F d ` µηψ c ` µ ηψ F ψ c ˙ ... “ det p ´ µψ F q det ˜ d ` µη ˜ ÿ n ě µ n ψ nF ¸ ψ c ¸ (2.7)Equation (2.6) and the next result imply formula (2.3). LU´IS DIOGO AND TOBIAS EKHOLM
Lemma 2.4.
After canceling critical points of ˆ φ near preimages of F ˘ “ f ´ p θ q in a µ -invariantway we get a function ¯ φ : Y K Ñ S such that the first determinant in (2.7) equals p ´ µ q timesthe exponentiated count of flow loops of ¯ φ , denoted ζ loop p µ q in (2.1) , and the second equals thedeterminant of the Morse–Novikov differential of ¯ φ , denoted τ Morse p µ q in (2.2) .Proof. Consider first the first determinant:(2.8) det p ´ µψ F q “ det exp p log p ` µψ F qq “ exp p tr log p ´ µψ F qq “ exp ˆ ´ ÿ n µ n tr p ψ nF q ˙ . To compare to (2.1), we need to relate tr p ψ nF q with the number of flow loops of homological degree n . Recall that the critical points in p C p F q and q C p F q cancel in pairs. The behavior of Morse flowsat such cancellations is well understood: if a cancels b along a shrinking flow line from a to b , thenrigid flow lines going into a are continued by the flow lines leaving b . It follows that a flow linecontributing to tr p ψ nF q closes to a flow loop going n times around a generator of H p ¯ Y K q .Not all flow loops in ζ loop come from ψ nF , though. There is a flow line from p C p F q to q C p F q (recall that F has a unique minimum), which contributesexp ˜ ÿ n ě µ n n ¸ “ ´ µ to ζ loop . This explains why we need to multiply (2.8) with p ´ µ q to obtain ζ loop .We now look at the second determinantdet ` d ` µη ` ` µψ F ` µ ψ F ` . . . ˘ ψ c ˘ . We need to argue that the matrix we are taking the determinant of can be identified with M p µ q in(2.2). The d term accounts for the ‘short’ flow lines in the Novikov differential, that do not pickany power of µ . The term ηψ nF ψ c can be interpreted as the flow lines obtained by starting withthe unstable manifold of an index 2 critical point ( ψ c ), canceling n intermediate Bott maxima andminima ( ψ nF ), and then intersecting with the stable manifold of an index 1 critical point ( η ). Thisgives the ‘long’ flow lines in the Novikov differential.Finally, we need to check that there are no flow loops and flow lines other than the ones countedin the formula. To see this, consider flow loops and flow lines that stretch between at most n Bott critical manifolds. As the Morse perturbation is turned off, it is standard that every flowline converges to a configuration as above and the corresponding finite dimensional gluing of flowlines gives a unique flow lines near each broken configuration for all sufficiently small perturbations.Our formulas are inductive limits in flow length, and taking smaller and smaller perturbations as n increases establishes the result. (cid:3) Flow loops, flow lines, and holomorphic curves
In this section we describe in more detail how a knot K has an associated conormal Lagrangian L K Ă T ˚ R diffeomorphic to S ˆ R and a ‘knot complement Lagrangian’ M K Ă T ˚ R diffeo-morphic to R z K , as well as non-exact deformations of these Lagrangians. We study holomorphicdisks in T ˚ R with boundary on L K or M K , and annuli and strips with one boundary componenton R and the other on L K or M K . For M K , we establish a 1-1 correspondence between theseholomorphic annuli and disks, and Morse flow loops and flow lines in R z K , respectively. Theproof uses [Ekh07] but avoids much of the complications there since our Lagrangian submanifoldshere have fronts without singularities. UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 9
Models of L K and M K . Let K Ă R be an oriented knot. We will consider exact and non-exact Lagrangians associated to K in T ˚ R , as in [AENV14, Section 6]. Let L K Ă T ˚ R denoteits Lagrangian conormal: L K “ tp q, p q P T ˚ R : q P K, p | T K “ u . Consider a small tubular neighborhood ν p K q Ă R of K , with projection π : ν p K q Ñ K . Let dθ bea constant 1-form on K of integral 1, and η L – π ˚ dθ . Given δ ą
0, write L δ K for the Lagrangianthat is obtained by applying to L K the non-exact symplectomorphism of T ˚ p ν p K qq that is givenby p q, p q ÞÑ p q, p ` δ η L p q qq . Note that L K X R “ K and L δK X R “ ∅ .Let M K denote the knot complement Lagrangian that is obtained by applying Lagrange surgeryalong the intersection L K X R “ K . Then M K is diffeomorphic to R z K . See [MW18] for adescription of Lagrangian surgery on a clean intersection, and for the fact that if the intersectionlocus is connected, then the result of surgery on two exact Lagrangians can also be made exact.Recall from Section 2.1 that f is an S -valued Morse function on the complement of a tubularneighborhood of the knot K Ă S . Using the tubular neighborhood, we extend f to S z K bymaking it approximately constant along rays in that neighborhood. Let dθ be a constant 1-formon S and η M “ f ˚ dθ . Note that ż m η M “ , where m is a meridian circle of K . Assume that S z R is not a critical point of f . When wepick a metric g and consider the vector field dual to η M , we will assume that has a neighborhoodthat does not contain periodic points of the gradient vector field, or points in gradient flow linesconnecting critical points of consecutive indices. Given δ ą
0, write M δ K for the Lagrangian that isobtained by applying to M K the non-exact symplectomorphism of T ˚ p R z K q given by p q, p q ÞÑ p q, p ` δ η M p q qq . Note that M δ K is graphical outside of a small neighborhood of K , namely it is the graph of the 1-form δ η M . Given our assumptions on f , we know that the 1-form η M is transverse to the 0-sectionand that it has no critical points of index 0 or 3. Remark . The non-exact Lagrangian shifts L δ K and M δ K do not have cylindrical ends, but theyare asymptoptic to a cylinder on Λ K . For a Lagrangian filling with cylindrical ends, R -translationinvariant moduli spaces of holomorphic disks at infinity form boundary components of correspond-ing moduli spaces in the filling. In particular such families can be continued as solution spaces ofdisks in the filling. The analogous fact for asymptotically cylindrical fillings is also true. To seethis, note that the non-exact perturbation is of fixed size whereas the symplectic form in the sym-plectization grows exponentially. Therefore, in sufficiently large disk bundles there is an arbitrarilysmall change of the almost complex structure (for instance, given by conjugation by a map taking aLagrangian to its δ -shift) making the original disks holomorphic with the right boundary condition.We now consider holomorphic disks with boundary on the Lagrangians L δK and M δK , possiblywith a puncture asymptotic to a Reeb chord. Since such punctured holomorphic disks are injectivenear the punctures, they are regular for a generic almost complex structure.Our next result shows that for sufficiently small δ ą
0, moduli spaces of once punctured holo-morphic disks are unaffected by the non-exactness of the Lagrangians L δ K and M δ K . Hence small There are two ways of performing Lagrange surgery, which might not be even isotopic. Nevertheless, the twoversions of M K induce the same augmentations (which could be seen from the relation between (cid:15) L K and (cid:15) M K mentioned in the proof of Lemma 4.22) below, so we do not need to distinguish them in this paper. non-exact shifts can also be used to compute the augmentations associated to L K and M K . Re-call that the energy of a holomorphic curve with compact domain Σ and Lagrangian boundary u : p Σ , B Σ q Ñ p T ˚ R , L q is E p u q “ ş Σ u ˚ ω . If B Σ has punctures, then the definition of energyof u must be adjusted and one uses the Hofer energy in the non-compact ends of the target, asin [BEH ` p iii q below is about generalized holomorphic disks, which are trees of finitelymany holomorphic disks with boundary on a Lagrangian, where the edges correspond to inter-sections with suitably chosen bounding chains. For details, see [AENV14, EN18]. In this paper,generalized holomorphic curves are not used in the theorems stated in Section 1. They appear onlyin Section 7. Lemma 3.2.
Let J be an almost complex structure on T ˚ R that is standard near the zero section.For any E ą , there is δ ą and a neighborhood J of J such that for any ă δ ď δ thefollowing holds for L δ “ L δ K or L δ “ M δ K and complex structures in J . p i q There are no non-constant closed holomorphic disks of energy less than or equal to E , withboundary on L δ . p ii q Assume that all moduli spaces of disks with one positive puncture on L are transversely cutout. If a is a Reeb chord of Λ K of degree 0, there is a natural 1-1 correspondence betweenholomorphic disks of energy less than or equal to E with one puncture asymptotic to a andwith boundary on L δ and L , respectively. (iii) For each a of degree 0, the space of generalized holomorphic disks on L δ with a positivepuncture asymptotic to a is canonically isomorphic to the corresponding space for L (whichis simply the space of ordinary holomorphic disks). It follows in particular that L δ and L define identical augmentations.Proof. Statement p i q is a consequence of Gromov compactness. If there is a non-constant holo-morphic disk u δ : p D , B D q Ñ p T ˚ R , L δ q , then r u δ | B D s P H p L δ ; Z q is non-trivial, since otherwise u δ would have vanishing area, by exactness of ω in T ˚ R . If p i q does not hold then there is asequence of such non-constant disks u δ , δ Ñ L , which contradicts exactness of L . The existence of sucha disk follows from the fact that the homotopy class of the boundary in a sequence of holomorphicdisks with Lagrangian boundary is preserved under Gromov limit, see [Fra08, Proposition 3.2(ii)].Statement p ii q is a consequence of compactness and transversality. Denote by M a,E p L δ q themoduli space of punctured disks for L δ in the statement. Consider the space M a,E pr , δ sq – ď δ Pr ,δ s M a,E p L δ q . By transversality, for L , for δ ą M a,E p L q is the boundaryof a component of the space M a,E pr , δ sq that is a compact 1-manifold and that projects as animmersion to r , δ s . Write M for the collection of all such components. We claim that thereexists 0 ă δ ă δ such that all disks in M a,E p L δ q for δ ă δ lie in M . This is again a consequenceof Gromov compactness.Statement p iii q follows from p i q and p ii q : generalized holomorphic disks are combinations of onedisk with positive puncture and several disks without punctures. Since there are no disks withoutpunctures and the disks with punctures are naturally identified as δ varies between 0 and δ , thestatement follows. (cid:3) Remark . We will see in Lemma 4.22 that, in the flow tree model for knot contact homology[EENS13] we can arrange that no Reeb chord of degree 0 is the asymptotic limit of a punctureddisk with boundary on L K . In this case, Gromov compactness suffices to imply that below anyenergy threshold, and for δ sufficiently small, L δ K bounds no holomorphic disks with one boundarypuncture asymptotic to a Reeb chord of degree 0. Put differently, the 1-1 correspondence in part p ii q of Lemma 3.2 is between empty sets. UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 11
Holomorphic annuli.
We consider next holomorphic annuli in T ˚ R between the zero sectionand the Lagrangians L δ K and M δ K , respectively.Given a real number R ą
0, call A R : “ r , R s ˆ S the annulus of modulus R (or of conformalratio R ). Given a Lagrangian L in T ˚ R , define M an p L q “ M an p R , L q : “ ď R Pp , (cid:32) u : A R Ñ T ˚ R : B J u “ , u pt u ˆ S q Ă R , u pt R u ˆ S q P L ( to be the space of J -holomorphic annuli of arbitrary modulus, from R to L . Given E ą
0, denoteby M E an p L q the subset of annuli of energy (that is, ω -area) at most E .The automorphism group of the Riemann surface A R consists of domain-rotations in the S -factor, and can be identified with S . Hence, S acts on M an p L q by pre-composition with domainautomorphisms. Let u P M an p L q be an annulus with multiplicity m p u q P Z ě . This means that u factors as v ˝ ϕ , where ϕ – A m p u q R Ñ A R is an m p u q -to-1 cover and v is an annulus that doesnot admit such a factorization. Then, the isotropy group of u under the S -action on M an p L q is Z {p m p u q Z q .Denote the quotient by Ă M an p L q – M an p L q{ S . Since isotropy groups of multiply covered annuli are non-trivial, this quotient space is an orbifoldrather than a manifold.Let us consider now the case where L is of the form L δ K . Since the Maslov class of theseLagrangians is zero, the expected dimension of the spaces M an p L δ K q of parametrized annuli is 1.Since L K is homotopy equivalent to K , we can write the homology class “ u | t R uˆ S ‰ “ d p u q x on H p L δK ; Z q , for some d p u q P Z , where x is the fundamental class of the oriented knot K . A simplecalculation shows that the energy of a holomorphic annulus u P M an p L δ K q of modulus R is E p u q “ ż A R u ˚ ω “ δ d p u q and since energy is non-negative and R X L δK “ ∅ , d p u q ą
0. Note that, given an integer d ą u P M δd an p L δ K q (of energy up to δd ) are exactly those with d p u q ď d .It will be useful to assume that K is a real analytic submanifold of R . By [CELN17, Lemma8.6], there is a compatible almost complex structure J on T ˚ R for which K has a neighborhood ν p K q that admits a holomorphic parametrization ϕ : S ˆ p´ , q ˆ B Ñ ν p K q , where B r Ă C denotes the ball of radius r ą ‚ K “ ϕ p S ˆ t u ˆ t uq , ‚ R X ν p K q “ ϕ p S ˆ t u ˆ B q and ‚ L K X ν p K q “ ϕ p S ˆ t u ˆ iB q ,where B r Ă R is the ball of radius r . Identifying the factor S ˆ p´ , q with a neighborhood ofthe zero section in T ˚ S , we can further assume that ‚ L δ K X ν p K q “ ϕ p S ˆ t δ u ˆ iB q for δ sufficiently small. Lemma 3.4.
For any d ą , there is δ ą such that for any ă δ ď δ there is a natural 1-1correspondence between covers of K of order up to d and elements in M δd an p L δ K q . All such annuliare regular for the chosen J .Proof. We begin by showing that, for fixed d ą δ ą u P M δd an p L δ K q are contained in the fixed neighborhood ν p K q . If we assume that u p p q R ν p K q , thenthere is a ball of radius around u p p q that intersects at most one of the Lagrangians R and L δ K .It follows by the monotonicity lemma with Lagrangian boundary conditions [CEL10, Lemma 3.4]that the area of u is at least some constant K ą
0. Since the area of such u goes to 0 as δ Ñ u lies inside ν p K q for δ sufficiently small. For such u , we consider the projection to B as in the definition of ϕ above, where L Y R mapsto R Y i R . By the exactness of R and L δ K restricted to B , this projection is constant. Thelemma is now reduced to studying holomorphic annuli on the target S ˆ p´ , q , with boundarycomponents on S ˆ t u and S ˆ t δ u , which are just unbranched covers of the annulus betweenthese two circles. An explicit computation shows that the linearized Cauchy–Riemann operatorsof these holomrphic annuli are surjective, yielding regularity. (cid:3) Lemma 3.4 implies that, for every fixed integer d ą δ sufficiently small, the connectedcomponents of M δd an p L δ K q are finitely many circles, each consisting of domain rotations of someannulus u of a certain multiplicity m p u q . The moduli space Ă M δd an p L δ K q is therefore a finite collectionof orbifold points, with isotropies given by Z {p m p u q Z q . We define the annulus counting functionsAn δ,dL K p λ q : “ ÿ r u sP Ă M δd an p L δ K q σ p u q m p u q λ d p u q P C r λ s , where σ p u q P t˘ u . This sign is defined as usual in holomorphic curve theory, via the Fukaya orien-tation [FOOO09] that is defined for curves with boundary by deforming the problem to a problemson disks with trivial boundary conditions and additional closed curves glued in as necessary. Herewe get the sign by deforming the annulus to an infinite annulus viewed as two disks, each withtrivial boundary condition, where the kernels of constant solutions are intersected upon gluing. Byour orientation conventions for conormals this intersection gives the orientation of the knot andhence the sign is ` u P M δd an p L δ K q . We also defineAn L K p λ q : “ lim d Ñ8 lim δ Ñ An δ,dL K p λ q P C rr λ ss . Remark . The count of annuli above can be compared to the count of holomorphic disks in[EKL18] and [EKL20], where so called basic disks are counted by the contribution of all theirmultiple covers. In the present case one could reformulate the count above by counting only simplycovered annuli, but counting them by the contributions of all their multiple covers. In other words,if M an p L δ K q is the moduli space of simple annuli thenAn L K p λ q : “ lim d Ñ8 lim δ Ñ ÿ r u sP Ă M dδ an p L δ K q 8 ÿ m “ σ m p u q m λ m d p u q P C rr λ ss . See Remark 2.1 above for another context where one also counts all the multiple covers associatedto a simple object.
Remark . The count of annuli can also be compared to the count of holomorphic curves inthe framed skein module of the brane as in [ES19], where only so called bare curves are counted.Here we would then have to perturb out the multiply covered annuli and count them in the U p q -skein module. One can then simplify the count and map the U p q framed skein to ‘homology andframing’ which corresponds to counting generalized holomorphic curves. Our count here shouldthen correspond to counting generalized holomorphic curves of Euler characteristic χ “
0, whichis a certain limit of the generalized curve count. In particular, the exact linking and self-linking ofthe boundary does not matter here since such terms would contribute only generalized curves oflower Euler characteristic. Since there are no holomorphic disks, the count Ψ L K , R of disconnectedgeneralized curves should then be of the formΨ L K , R “ exp p An L K p λ q ` O p g s qq . Lemma 3.4 has the following consequence, which will be useful below.
UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 13
Corollary 3.7.
For any knot K Ă R we have An L K p λ q “ ÿ k ą λ k k “ ´ log p ´ λ q . We now turn to holomorphic annuli between R and M δ K . Just like L δ K , the Lagrangian M δ K hasMaslov class 0 and hence the expected dimension of the space M an p M δ K q of parametrized annuli is1. Recall the closed 1-form η M “ f ˚ dθ on R z K chosen above. Fix a Riemannian metric g on R and let η ˚ M denote the vector field that is the metric dual of η M . Given a flow loop or a flow line γ Ă R of η M , we define its action as a p γ q “ ż γ η M . Recall that, for a generic pair p f, g q , the critical points and flow loops of η ˚ M are hyperbolic, andthe intersections of stable and unstable manifolds are transverse. We will assume that p f, g q is ageneric pair in this sense.We will next adapt the metric and the Lagrangian M K to the rigid flow loops and lines as in[Ekh07, Section 4.3]. The modifications here correspond to the most basic cases in that paper,since our flow objects here are simply flow lines (there is no branching as in flow trees). We deformthe Lagrangian and the metric so that the metric is flat, the gradient is constant along a flow loop,and of the form in [Ekh07, Section 4.3.6–7] near a flow line. The construction here depends on theaction level a . As we increase the action, new flow objects need to be taken into account and weshrink the neighborhoods where the Lagrangian and metric in previous steps were normalized. Wedo that in such a way that when passing from action level a to a , a ą a , we keep the metricand Lagrangian unchanged in some subset of the set where it was previously normalized.Fix the natural almost complex structure J on T ˚ R determined by the metric g , see [Ekh07,Section 4.4]. Lemma 3.8.
For any a ą there exists δ ą such that, for any δ ă δ , there is a natural 1-1correspondence between rigid flow loops and flow lines of η ˚ M of action ă a , and J -holomorphicannuli and J -holomorphic strips, respectively, between R and M δ K of action ă δ a . More precisely,there exists a neighborhood of any simple rigid flow loop that contains exactly one simple rigidholomorphic annulus, and there exists a neighborhood of any rigid flow line that contains exactlyone holomorphic strip. Moreover, there is a natural 1-1 correspondence between multiple flow loopsover simple loops and multiply covered annuli over simple annuli. These J -holomorphic annuli andstrips are regular.Proof. The proof is a simpler version of the corresponding result in [Ekh07]. The case of stripsfollows immediately from [Ekh07, Theorem 1.1].Consider the case of annuli. We first show that rigid annuli converge to rigid flow loops as δ Ñ δ times the actionas in [Ekh07, Lemma 5.2 and 5.4]. Then one uses subharmonicity of the square of the momentumcoordinate to confine the holomorphic curves to an O p δ q -neighborhood of the zero section as in[Ekh07, Lemma 5.5]. With this established, basic elliptic estimates give control of the C -normof the holomorphic maps as in [Ekh07, Lemma 5.6]. (Compared to the flow tree case, only thesimplest case is required here.) At this stage [Ekh07, Lemmas 5.13 and 5.18] gives the desiredconvergence to flow loops.We next show that there is a unique holomorphic annulus near each flow loop. Our choice offlat metric near the flow loop and the deformation of the Lagrangian gives an obvious holomorphicannulus over the flow loop. An explicit calculation shows that the linearized operator is uniformlysurjective, and it follows from this transversality that the annulus is unique in a small neighborhood. (cid:3) Remark . The proof above works for the shift by a 1-form of any graphical exact Lagrangian,since no flow trees need to be considered. In particular, this argument could replace the proof ofLemma 3.4 above.We can now define generating functions An δ,dM K p µ q and An M K p µ q in a manner analogous to thedefinitions given above for L K . The definition includes again σ p u q P t˘ u . We have σ p u q “ σ p γ q ,where γ is the flow loop corresponding to u by the previous lemma, and σ p γ q is as in equation(2.1). Proof of Theorem 1.3.
The result follows from combining equation (2.3) with Lemma 3.8. (cid:3)
Remark . The product ζ an p µ q ¨ τ str p µ q in Theorem 1.3 and the corresponding λ -dependentanalogue for L K can be thought of as torsions of the Lagrangian Floer complexes CF ˚ p R , M δ K q and CF ˚ p R , L δ K q , respectively (where M δ K and L δ K are twisted by local systems). The invarianceof Floer torsion under change of almost complex structure and Hamiltonian isotopy is studied in[Lee03]. We also discuss the invariance of Floer torsion in Section 7.4. Knot contact homology, augmentations, and linearized homology computations
In this section we first review knot contact homology, the Chekanov-Eliashberg dg-algbera of theLegendrian conormal of a knot generated by Reeb chord and with differential counting puncturedholomorphic disks, including a description of additional geometric data that is used to define thedifferential. We then turn to augmentations of knot contact homology and associated augmentationvarieties. Finally, we compute the linearized contact homology associated to augmentations inducedby a class of Lagrangian fillings, by relating Reeb chords to geodesics and the space of paths startingand ending on the knot. The result of the calculation gives structural results for augmentationvarieties of the dg-algebras of Legendrian conormals of knots.4.1.
Geometry of coefficients in the dg-algebra of knot contact homology.
The dg-algebraof an oriented knot K Ă R is the Chekanov–Eliashberg differential graded algebra of its Legendrianconormal Λ K Ă ST ˚ R . We denote it A K . It is a tensor algebra freely generated by the Reeb chordsof Λ K . The differential counts punctured holomorphic disks in R ˆ ST ˚ R , with boundary compo-nents mapping to R ˆ Λ K and with boundary punctures (one positive and arbitrarily many negative)asymptotic to Reeb chords. The coefficient ring of A K is the group ring C r H p ST ˚ R , Λ K ; Z qs . Thiscoefficient ring can be identified with a Laurent polynomial ring R – C r λ ˘ , µ ˘ , Q ˘ s , once wemake the following choices. ‚ For each Reeb chord c , choose a capping half-disk v c : D ` Ñ ST ˚ R , where D ` is theintersection of the unit disk in the complex plane and the upper half plane, such that v c | D ` X R is a parametrization of c and where v c p D ` X S q Ă Λ K , where R is the real lineand S is the unit circle. Up to homotopy, two capping half-disks differ by an element in π p ST ˚ R , Λ K q . ‚ We fix embedded curves x and p in Λ K that generate H p Λ K ; Z q and with intersectionnumber x ¨ p “ x is null-homologous in M K « R z K andthat p is null-homologous in L K « S ˆ R . We also require that the projection of x to K preserve orientation. This fixes their classes in H p Λ K ; Z q . We call x the longitude and p the meridian , respectively, and we sometimes identify them with the homology classes thatthey represent.As we shall see, the formulas in Theorem 1.2 are still valid if we change the meridiancurve by a transformation of the form(4.1) p x, p q ÞÑ p x, p ` kx q for some k P Z . We call this a change of framing . UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 15 γ ´ a Σ x ´ b Σ p capping uS p γ q γ disksΣ p γ q u Figure 2.
Completing u into a closed surface to determine the Q -power ‚ Choose a splitting of the short exact sequence(4.2) 0 H p ST ˚ R ; Z q H p ST ˚ R , Λ K ; Z q H p Λ K ; Z q , as indicated by the dashed line. The sequence starts with 0 because the fundamental classof Λ K vanishes in H p ST ˚ R ; Z q by the Gauss-Bonnet theorem. Let t P H p ST ˚ R ; Z q bethe class of a generator, we think of it as a fiber of the S -bundle ST ˚ R Ñ R . This classis unique if we require the intersection with a section of the bundle to be `
1. The splittingcan be thought of as a choice of two surfaces in ST ˚ R , one having its boundary componenton x and the other on p . Denote these surfaces by Σ x and Σ p , respectively. Note that thesplitting is well-defined up to adding integer multiples of t to Σ x and Σ p . We also pick asection s of the trivial S -bundle ST ˚ R Ñ R that is generic in the following sense. Thegraph of s is disjoint from all Reeb chords c and is transverse to all capping half-disks v c ,to the capping surfaces Σ x and Σ p , and to Λ K . A class in H p ST ˚ R q is determined by itsintersection with the graph of s .We will write λ “ e x , µ “ e p and Q “ e t for the group ring variables corresponding to the homol-ogy generators indicated above. We describe how to compute the R -coefficient of the contributionfrom a holomorphic curve u , transverse to the section s , to the differential in A K , in accordancewith the choices that we made.Consider first the λ and µ powers: use the capping disks associated to the asymptotic chords of u to produce a closed disk u that then represents an element in H p ST ˚ R , Λ K ; Z q . The boundary γ of u gives a class in H p Λ K ; Z q , which can be written in the form r ax ` bp s , where a is theintersection number γ ¨ p and b is x ¨ γ . The λ, µ coefficients of the contribution of u will then be λ a µ b .Consider next the Q power: the closed disk u gives an element of H p ST ˚ R , Λ K ; Z q , and theboundary γ of u is homologous to γ – ax ` bp . Pick a 2-dimensional oriented surface with boundary S p γ q and a map S p γ q Ñ Λ K , such that the boundary of S p γ q maps to γ ´ γ . We then have aclosed surface in ST ˚ R by concatenating u with the image of S p γ q and with a copies of ´ Σ x and b copies of ´ Σ p (where the sign means reverse orientation). See Figure 2. The Q -power of u in A K is the intersection number of this closed surface with the image of the section s . Note that r Λ K s “ P H p ST ˚ R ; Z q implies that this intersection number is independent of the choice ofcobordism S p γ q as different choices differ by a multiple of the fundamental class of Λ K . Remark . The change of framing (4.1) above induces a change of variables in R of the form p λ, µ, Q q ÞÑ p λ, λ k µ, Q q , for some k P Z , and a change of splitting (4.2) induces p λ, µ, Q q ÞÑp λQ k , µQ l , Q q , for some k, l P Z . For some steps in the derivation of the formula in Theorem 1.2, it will be important to understand Q -powers in more detail. We discuss this next, the result is summarized in Lemma 4.2 below.Consider first how the Q -power depends on the choices of capping surfaces Σ x , Σ p . These choicesaffect the intersection with the section s of the surface Σ p γ q , which is the concatenation of thecobordism S p γ q with ´ a Σ x ´ b Σ p in Figure 2. Recall that a “ γ ¨ p and b “ x ¨ γ . Note thatΣ p γ q ¨ s , thought of as a function of γ , is well-defined on H p Λ K z s ; Z q . The capping surface Σ x canbe changed by taking a connected sum with an S -fiber, and this changes Σ p γ q ¨ s by subtracting a . Similarly, changing Σ p to Σ p S subtracts b from Σ p γ q ¨ s . Therefore, we can think of Σ p γ q ¨ s as a family of homomorphisms Φ Σ x , Σ p : H p Λ K z s ; Z q Ñ Z such that, if we fix one element Φ , we can obtain all other elements by adding homomorphisms ofthe form γ ÞÑ mγ ¨ x ` nγ ¨ p for some p m, n q P Z . In other words, the collection of homomorphismscan be thought of as a torsor over H p Λ K ; Z q .Let us now consider an alternative way of computing the powers of Q associated to u , by ex-pressing the intersection Σ p γ q ¨ s differently as follows. This will be used later in the paper. Recallthat Λ K is null-homologous in ST ˚ R , and intersects s transversely. Hence, this zero dimensionalintersection has signed count equal to zero, and is the boundary of a 1-dimensional submanifold τ Ă Λ K . On H p Λ K ; Z q , τ is determined up to connected sum with x or p . We can thus defineanother family of homomorphisms Ψ τ : H p Λ K z s ; Z q Ñ Z by Ψ τ p γ q “ γ ¨ τ , where the intersection is now taken in Λ K . Observe that Ψ τ x p γ q “ Ψ τ p γ q ` γ ¨ x and Ψ τ p p γ q “ Ψ τ p γ q ` γ ¨ p . If we show that Φ Σ x , Σ p “ Ψ τ for one choice of Σ x , Σ p and τ , thenwe can conclude that the collection of all Φ Σ x , Σ p agrees with the collection of all Ψ τ , and can thusreplace an intersection Σ p γ q ¨ s with γ ¨ τ for an appropriate τ .Fix now a choice of Σ x , Σ p such that Σ x ¨ s “ Σ p ¨ s “ S guarantees the existence of such choices). Fix also a choice of τ such that τ ¨ x “ τ ¨ p “ x and p ). Then, indicating by a subscript whereintersections take place, we have ` Σ p γ q ¨ s ˘ ST ˚ R “ ` S p γ q ¨ s ˘ ST ˚ R “ ` S p γ q ¨ p s X Λ K q ˘ Λ K “ ` γ ¨ τ ˘ Λ K , where the last identity follows from the usual argument showing that the linking number of twosubmanifolds of Euclidean space can be computed by intersecting either submanifold with a sub-manifold bounding the other. This shows the following. Lemma 4.2.
Given choices of capping half-disks and capping surfaces Σ x , Σ p , the exponent of Q associated to a holomorphic curve u is given by ` u ¨ s q ST ˚ R ` ` γ p u q ¨ τ q Λ K for a suitable choice of 1-dimensional submanifold τ Ă Λ K above. Here, γ p u q is the boundary ofthe disk u . Different choices of Σ x , Σ p correspond bijectively to different choices of τ . (cid:3) Remark . We can think of the intersection Σ p γ q ¨ s as a version of a linking number between γ and s in ST ˚ R . The discussion above shows that this can be replaced with γ ¨ τ , which can beinterpreted as a linking number between γ and s X Λ K in Λ K .4.2. The augmentation variety and the augmentation polynomial.
In this section wediscuss various aspects of augmentations. Consider, as in Section 4.1, A K with coefficients in R “ C r λ ˘ , µ ˘ , Q ˘ s . An augmentation for A K is a unital chain map of dg-algebras, (cid:15) : A K Ñ C , UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 17 where C is supported in degree 0 and the chain map condition means that (cid:15) ˝ B “
0. The linearizedcontact homology with respect to (cid:15) , denoted
KCH (cid:15) ˚ p K ; C q , is the homology of the C -vector space KCC (cid:15) ˚ p K ; C q generated by Reeb chords of Λ K , with the following differential. Given a chord c ,interpret its knot contact homology differential B c as a polynomial in variables corresponding tothe Reeb chords of Λ K . To find the coefficient of a chord b in the linearized differential of c , takethe partial derivative of that polynomial with respect to the variable b and apply (cid:15) to the resultof that partial differentiation. Equivalently, for each non-constant monomial on chords in B c , sumthe result of applying (cid:15) to the coefficient variables λ , µ and Q , and to all but one of the chords inthe monomial, in all possible ways.An exact Lagrangian filling L of Λ K , inside an exact symplectic filling p X, dα q of ST ˚ R , inducesaugmentations as follows. The augmentation associates to every degree 0 Reeb chord a of Λ K thecount of holomorphic disks in Y , with boundary on L and one positive puncture asymptotic to a .The values of an augmentation on elements of the coefficient ring R must be compatible with theinclusion maps on homology:(4.3) 0 H p ST ˚ R ; Z q H p ST ˚ R , Λ K ; Z q H p Λ K ; Z q H p L ; Z q H p X ; Z q H p X, L ; Z q H p L ; Z q . Example . For every knot K , the conormal Lagrangian L K in T ˚ R is an exact Lagrangian fillingof Λ K . The meridian p is null-homologous in L K and the generator t P H p ST ˚ R ; Z q vanishes in H p T ˚ R ; Z q “
0. There is an associated 1-parameter family of augmentations, that we denote by (cid:15) L K , assigning 1 to µ and Q and an arbitrary element in C ˚ to λ . Similarly, the exact Lagrangianfilling M K , obtained from clean intersection surgery on L K and R , on which the longitude x isnull-homologous, gives a family of augmentations (cid:15) M K , assigning 1 to λ and Q and an arbitraryelement in C ˚ to µ .It is useful to think of the collection of all augmentations of A K geometrically. To this end, weuse the following notation. Label the Reeb chords of degree 0 by a i and denote the ordered setof such chords a “ p a i q . Write C a for a vector space with basis a and write ˜ R – R r a s for thepolynomial ring in variables a . Let ˜ I K Ă ˜ R be the ideal generated by the differentials of chords ofdegree 1 in A K . We will use the following geometric objects in connection with augmentations: Definition 4.5. ‚ The full augmentation scheme of K is the affine scheme ˜ V K – Spec ´ ˜ R { ˜ I K ¯ . ‚ The full augmentation variety is the algebraic set ˜ V K – V p ˜ I K q “ (cid:32) p (cid:15) p λ q , (cid:15) p µ q , (cid:15) p Q q , (cid:15) p a qq P p C ˚ q ˆ C a | (cid:15) is an augmentation ( . This is the set of closed points in ˜ V K . ‚ The augmentation variety V K of K is the union of maximal-dimensional components of theZariski-closure of the set (cid:32) p (cid:15) p λ q , (cid:15) p µ q , (cid:15) p Q qq P p C ˚ q | (cid:15) is an augmentation for A K ( “ π p ˜ V K q , where π : p C ˚ q ˆ C a Ñ p C ˚ q is the projection that forgets the Reeb chord variables a i . ‚ Corollary 4.12 below shows that I p V K q is a principal ideal. A generating element of I p V K q is called an augmentation polynomial of K , and denoted Aug K p λ, µ, Q q .Remark K ) . Often Aug K can be obtained using elimination theory, as follows.If I K – R X ˜ I K , then V K Ă Cl p π p ˜ V K qq Ă V p I K q , where Cl denotes Zariski-closure. Hence, I p V K q Ą I p V p I K qq “ ? I K , by Hilbert’s Nullstellensatz. Therefore, if I K is not the zero ideal in R ,then Aug K ‰ V K Ă p C ˚ q is either 2- or 3-dimensional. Hence, V K iscontained in the top dimensional component of V p I K q , which we denote by X K Ă p C ˚ q . Since thedimension of X K is 2 or 3, each irreducible component of X K is cut out by an irreducible polynomial(see for instance [Har77, Chapter I, Proposition 1.13] ) , and hence I p X K q is a principal ideal. If f is a generator of this ideal, then Aug K divides f .The polynomial f can in principle be explicitly computed as follows. From a Gr¨obner basis for˜ I K , one can obtain a Gr¨obner basis for I K ([CLO15, Chapter 3 §
1, Theorem 2]). Then one can findgenerators for ? I K ([CLO15, page 184]) and obtain a minimal decomposition of ? I K ([CLO15,page 218]). A generator f of I p X K q can be obtained from such a decomposition. If we knew that V K “ V p I K q (or equivalently, that I p V K q “ ? I K ), then Aug K “ f .In case we know that I K is a principal ideal, then there is a potentially simpler computationalapproach to determining Aug K : I K is a principal ideal if and only if its reduced Gr¨obner basis hasa single element (which is a generator g ).Then a generator f of ? I K would be given by a reduction of g , according to [CLO15, Chapter 4 §
2, Proposition 9] (if g “ ug a . . . g a r r is a factorization into irreducible polynomials, then a reductionof g is g . . . g r ). As before, we could say that Aug K divides f , and that they would coincide if weknew that I p V K q “ ? I K . Example . The augmentation polynomial of the unknot is 1 ´ λ ´ µ ` λµQ , see [AENV14]. Example . Example 4.4 (together with Proposition 4.11 below) implies that, for every K , V K contains the points p λ, , q and p , µ, q , for all λ and µ P C ˚ .We next consider first order properties of the space of augmentations. We start with a generalfact, see [EN18, Remark 4.8]. Lemma 4.9.
Let (cid:15) be an augmentation of A K , and denote the corresponding closed point in ˜ V K also by (cid:15) . The Zariski tangent space to π ´ p π p (cid:15) qq Ă ˜ V K at the point (cid:15) is isomorphic to the degreezero (cid:15) -linearized homology KCH (cid:15) p K ; C q .Proof. We can assume that
KCC ˚ is supported in degrees 0, 1 and 2, as in [EENS13]. Denote theset of Reeb chords of degree 0 by a “ p a i q and the set of chords of degree 1 by b “ p b j q . Denote by C a and C b the vector spaces with bases a and b , respectively. Write Φ j for the contact homologydifferential db j , viewed as a polynomial in λ ˘ , µ ˘ , Q ˘ , and a i .The Zariski tangent space at (cid:15) P C a is given by the intersection of the kernels of the differentials Ş j ker p d (cid:15) Φ j q , where(4.4) d (cid:15) Φ j “ ÿ i B Φ j B a i ˇˇˇˇ (cid:15) da i , Noting that d (cid:15) Φ j is the co-vector dual to the (cid:15) -linearized contact homology differential Ψ (cid:15)j “ B (cid:15) p b j q of b j , we find, writing Ψ (cid:15) “ im B (cid:15) Ă C a for the subspace spanned by the Ψ (cid:15)j , that č j ker p d (cid:15) Φ j q – C a { Ψ (cid:15) “ KCH (cid:15) p K ; C q , where the last identity uses the fact that KCC ˚ is supported in non-negative degrees. (cid:3) It will be useful below to have a similar description of the whole tangent space to a point in˜ V K , instead of just the tangent spaces to the fibers of the projection π . To this end, we definethe full linearized contact homology { KCH (cid:15) ˚ p K ; C q as the homology of the chain complex obtained although the result is stated in [Har77] for subvarieties of affine space C n , the proof also holds in p C ˚ q n UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 19 by linearizing the contact homology DGA of K also with respect to the coefficients λ , µ , and Q .More precisely, this means that the linearized differential of b j is the dual vector in C ‘ C a of theco-vector which generalizes (4.4):(4.5) p d Φ j “ B Φ j B λ ˇˇˇˇ (cid:15) dλ ` B Φ j B µ ˇˇˇˇ (cid:15) dµ ` B Φ j B Q ˇˇˇˇ (cid:15) dQ ` ÿ j B Φ j B a i ˇˇˇˇ (cid:15) da i We then have the following:
Lemma 4.10.
Let (cid:15) be an augmentation of A K . The Zariski tangent space to ˜ V K at (cid:15) is isomorphicto { KCH (cid:15) p K ; C q .Proof. Identical to the proof of Lemma 4.9. (cid:3)
Linearized contact homology computations and the augmentation variety.
The goalof this section is to establish basic properties of augmentation varieties. We first state the resultsand discuss immediate consequences, and later turn to the proofs. The first result is the following.
Proposition 4.11.
Let K Ă R be a knot. The full augmentation variety ˜ V K is a smooth two-dimensional complex algebraic variety in a Zariski-open neighborhood of the point p λ, µ, Q, a q “p , , , q . There is a Euclidean neighborhood U of this point in which ˜ V K admits a holomorphicparametrization by the coordinates p λ ´ , µ ´ q . We write p , , , q “ p , q and denote the corresponding augmentation by (cid:15) . Proposition 4.11has the following consequence: Corollary 4.12.
For every K , the augmentation polynomial Aug K exists (but might be 0).Proof. Since ˜ V K is two-dimensional and parameterized by p λ, µ q near p , q , the maximal dimen-sional components of Cl ´ π p ˜ V K q ¯ Ă p C ˚ q are at least 2-dimensional. If they are 3-dimensional,then Aug K “
0. If they are 2-dimensional, then their union is cut out by a single polynomial (seefor instance [Har77, Chapter I, Proposition 1.13]). (cid:3)
Remark . Corollary 4.12 can be decuced also from results on the cord algebra in [CELN17] incombination with [AENV14, Section 6.11], see also [Cor17].
Remark . This is compatible with [Ng14, Conjecture 5.3], according to which Aug K “ I K “ R X ˜ I K is not the zero ideal in R , thenAug K ‰ p , q . Proposition 4.15. (1)
There is a Euclidean neighborhood U of p , q such that, for every augmentation (cid:15) P U ,we have KCH (cid:15)k p K ; C q – C if k “ or otherwise . (2) Let U be as in Proposition 4.11, and denote the parametrization of ˜ V K in U X U by (cid:15) λ,µ .Then there is a linear combination y p λ, µ q of Reeb chords, with coefficients analytic in λ and µ , representing a generator of KCH (cid:15) λ,µ p K ; C q . Write F p λ, µ, Q q – Bp y p λ, µ qq| a ÞÑ (cid:15) λ,µ p a q , where B denotes the differential in A K and a “ p a , . . . , a n q denotes the ordered set of allindex zero Reeb chords. Then, (4.6) pB Q F q| λ “ µ “ Q “ ‰ and (4.7) pB λ F q λ “ Q “ is non-constant with isolated zeros, as a function of µ. Propositions 4.11 and 4.15 imply Theorem 1.1.
Remark . In [Ng08, Proposition 4.4], it is shown that there is an isomorphism of Q r µ ˘ s -modules KCH (cid:15) MK p K ; Q q – p H p ˜ M K ; Q q ‘ Q r µ ˘ sq b Q r µ ˘ s p H p ˜ M K ; Q q ‘ Q r µ ˘ sq ‘ p Q r µ ˘ sq m , for some m ě
0, where ˜ M K is the universal Abelian cover of the knot complement. This alreadyimplies that the Alexander polynomial can be obtained from linearized knot contact homology.Proposition 4.15(1) implies that m “ K .4.3.1. A 1-parameter family of generators of linearized homology.
We begin by recalling some re-sults from [EN18]. Consider the 1-parameter augmentations (cid:15) L K induced by L K , parameterizedby λ P C ˚ , which can be thought of as the monodromy around K of a flat connection on a trivialcomplex line bundle over L K . Let KCC (cid:15)
LK,λ ˚ p K ; C q denote the chain complex of knot contacthomology, linearized with respect to this augmentation for a specific value of λ .Let also P K denote the space of paths in R that start and end in K , and by P K the space ofconstant such paths. There is a local coefficient system over P K , associated to the concatenationof group homomorphisms(4.8) π p P K q Ñ π p K q ˆ π p K q Ñ π p K q λ Ñ C ˚ , where the first map is induced by evaluation at the path endpoints, the second map is p g , g q ÞÑ g g ´ in the group, and the last map is induced by the monodromy λ of the flat connection. Let C λ ˚ p P K ; C q and C λ ˚ p P K ; C q be the corresponding singular chain complexes with local coefficients.Since the fiber of the endpoints map P K Ñ K ˆ K is the based loop space of R , the map gives aquasi-isomorphism C λ ˚ p P K ; C q Ñ C λ ˚ p K ˆ K ; C q (for the local system on K ˆ K that is also inducedby the sequence (4.8)). By [EN18, Lemma 4.5], there is a quasi-isomorphism(4.9) Θ λ : KCC (cid:15)
LK,λ ˚ p K ; C q Ñ C λ ˚ p P K , P K ; C q , defined by counting half-infinite holomorphic strips in T ˚ R with boundary in L K Y R . Indeed,using the action filtration and a suitable geodesic Morse model for C λ ˚ p P K , P K ; C q , generators ofthe two complexes can be identified and the map Θ λ is upper triangular with ˘ KCH (cid:15)
LK,λ k p K ; C q – H λk p P K , P K ; C q – C if k “ ,
20 otherwisefor any λ ‰ λ ‰
1, but the argument can also be adaptedto λ “ P K and the diagonal ∆ Ă K ˆ K ,and the connecting homomorphisms in the quotient exact sequence are maps δ : H λk p P K , P K ; C q Ñ H λk ´ p P K ; C q . An explicit calculation shows that δ is multiplication by 1 ´ λ on H . More precisely,take the Morse model for C λ ˚ p K ˆ K ; C q with one minimum ξ and one maximum ξ on ∆ givingthe minimum and a saddle point in K ˆ K , and an additonal saddle point η and a maximum η .For any relative cycle γ in C λ p P K , P K ; C q taken by the endpoints map to the degree one generator η in C λ p K ˆ K, ∆; C q , we have(4.11) δ p γ q “ p ´ λ q ξ . UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 21 L K y p λ q L K y p λ q a i L K a j R ˆ Λ K K x Y Figure 3.
The boundary of the moduli space in the proof of Lemma 4.18. In theconfiguration on the right, there could be arbitrarily many punctures capped bydisks with boundary in L K .We next note that if c is a cycle in KCC (cid:15)
LK,λ ˚ then Θ λ p c q is a relative cycle and we get δ pr Θ λ p c qsq P H λ ˚´ p P K ; C q . It turns out that the composition δ ˝ r Θ λ s is induced by a chain mapΦ λ : KCC (cid:15)
LK,λ ˚ p K ; C q Ñ C λ ˚´ p P K ; C q (i.e., for cycles c , δ p Θ λ p c qq “ Φ λ p c q ), that can be described as follows. The map Φ λ countsholomorphic disks in T ˚ R with boundary in L K and one boundary puncture asymptotic to a Reebchord of Λ K . If the Reeb chord is c with | c | “
1, then Φ λ p c q counts disks with a marked point onthe boundary that maps to K . If the Reeb chord is b with | b | “
2, then Φ λ p b q counts disks witha marked point on the boundary mapping to ξ P K . (This map is denoted d (cid:15) in [EN18, Lemma4.6].)Consider now the differential B λ – B (cid:15) LK,λ as a collection of λ -dependent linear maps. Note thatthe matrix of B λ has entries that are analytic in λ . It then follows from (4.10) that the kernelof B λ is an analytically varying field Z λ Ă KCC λ p K ; C q of subspaces and that the image of B λ is a analytically varying codimension one subspace B λ Ă Z λ . Noting that the map Φ λ dependsanalytically on λ , we find an analytic section y p λ q of Z λ { B λ near λ “ λ p y p λ qq “p ´ λ q ξ . Definition 4.17.
Let (4.12) G p λ, µ, Q q – Bp y p λ qq| a ÞÑ (cid:15) LK,λ p a q , where B denotes the differential in the DGA A K , viewed as a Laurent polynomial in p λ, µ, Q q anda polynomial in the degree zero Reeb chords a , and where we substitute each degree zero Reeb chord a i with (cid:15) L K ,λ p a i q , which is a function of λ . Lemma 4.18. Φ λ p y p λ qq “ pB µ G q| µ “ Q “ ¨ ξ Proof.
Recall from Section 4.1 that the µ -powers in the differential of A K are given by countingintersections of the boundaries of holomorphic disks with a longitude curve x in Λ K .Pick a non-compact surface Σ Ă L K , with a single boundary component going once around K and with boundary at infinity in the curve x .The moduli space of holomorphic disks in T ˚ R with boundary on L K , with one boundarypuncture asymptotic to y p λ q and one boundary intersection with Σ, is 1-dimensional, with a naturalcompactification given by adding the two boundary configurations in Figure 3. The boundaryconfiguration on the left contributes to Φ λ p y p λ qq . The configuration on the right contributes to pB µ G q| µ “ Q “ ¨ ξ . The lemma follows. (cid:3) Since y p λ q depends analytically on λ near λ “
1, we have that G p λ, µ, Q q is an analytic functionof p λ, µ, Q q in a neighborhood of p , , q . We next show that the function B µ G is non-constant(and therefore has isolated zeros) along the ‘ µ -axis’ given by t λ “ “ Q u . Corollary 4.19.
We have B µ G p , , q “ and B λ B µ G p , , q ‰ . Hence pB λ G qp , µ, q is non-constant with isolated zeros.Proof. Lemma 4.18, combined with the definition of y p λ q , implies that B µ G p λ, , q “ p ´ λ q .Taking the λ -derivative at 1 then proves the lemma. (cid:3) Next, we get more information about the function G , by using counts of holomorphic annuli in T ˚ R between R and L δK . For now, we will focus on the key points of the proof without goinginto analytical technicalities. The full treatment and a complete proof appear in Section 5. Proposition 4.20.
The following equation holds: λ pB µ G qp λ, , q “ p λ ´ qpB Q G qp λ, , q .Proof. Consider the 1-dimensional moduli space M an p y p λ qq of holomorphic annuli between L δK and R , with one boundary puncture on L δK asymptotic to y p λ q . By Lemma 5.4 below, the boundaryof this space consists of once punctured disks at infinity which intersect the bounding chain of arigid annulus, once punctured disks at infinity which intersect the 4-chain given by the graph ofthe vector field s , and disks at infinity with a negative puncture where a once punctured annulusis attached (see Figure 6). The last term vanishes since y p λ q is a cycle, the first term is given by B µ G p λ, , q ¨ ´ λ B λ An L δK p λ q ¯ “ B µ G p λ, , q ¨ λ ´ λ , where we use Corollary 3.7. The middle term is B Q G p λ, , q . The result follows. (cid:3) Corollary 4.21.
The Q -variation of G at p , , q does not non-vanish: pB Q G qp , , q ‰ .Proof. Differentiate the equality in Proposition 4.20 with respect to λ : B G B µ p λ, , q ` λ B G B λ B µ p λ, , q “ B G B Q p λ, , q ` p λ ´ q B G B λ B Q p λ, , q . Evaluate at λ “ (cid:3) Augmentations and full linearized homology.
We next show that for any knot K , there is amodel of the DGA A K for which the augmentations induced by L K and M K at p λ, µ, Q q “ p , , q act trivially on all degree zero Reeb chords. To this end, we represent K as a braid with n strandsaround the unknot. Then as explained in [EENS13] there are n p n ´ q degree zero generators of A K that we denote a i , and write a “ p a i q as usual. Lemma 4.22. If K is braided around the unknot, then the family of augmentations (cid:15) L K consistsof the points p λ, , , q , for arbitrary λ P C ˚ . This family intersects the 1-parameter family ofaugmentations (cid:15) M K (which lies over the set tp , µ, qu Ă V K ) at the point p , , , q .Proof. Denote the round unknot by U . The Legendrian Λ U does not have any Reeb chords ofdegree 0. Also, since L U is exact, there are no unpuctured holomorphic disks with boundary on L U .Consider degenerating the braided K to U . As K Ñ U , L K Ñ L U and the limit of an aug-mentation disk on L K is a holomorphic disk on L U with flow trees attached, see [EENS13]. As K approaches U , L K limits to a cover of L U without singularities, i.e., as a local diffeomorphism.Therefore there are no flow trees corresponding to disks with one positive puncture at a degree0 chord, and also no disks with flow trees attached since there are no disks on L U . It follows that (cid:15) L K acts trivially on the a i .Consider next the augmentation (cid:15) M K . As we degenerate M K to L K Y R , augmentation diskslimit to disks with boundary on L K or disks with switching boundary conditions as in [CELN17]. UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 23
As shown above, there are no disks with boundary on L K . In [AENV14], the moduli space of disksfor small smoothings near a rigid disk with switching boundary condition was described, see also[DRET18, Lemma 4.1] for a more detailed description. The result is that a corner switching from L K to R has a unique smoothing, and a corner from R to L K has two smoothings with boundariesthat differ by the meridian. This means that(4.13) (cid:15) M K “ (cid:15) L K ` m ÿ k “ ˘p ´ µ q k φ kL K Y R , where φ kL K Y R is a count of rigid disks with switching boundary conditions in L K Y R and 2 k corners. Therefore, at µ “ (cid:15) M K p a i q “
0, as claimed. (cid:3)
Remark . The argument in the proof of Lemma 4.22 applies more generally to conormals andassociated Lagrangian knot complements for knots in arbitrary position (not necessarily braidedaround the unknot). It implies that if M K is obtained from surgery on L K Y R , then for sufficientlysmall surgery parameter formula (4.13) holds. Remark . The values of the augmentations (cid:15) M K on chords of degree 0 are written in [Ng08,Definition 4.6], using slightly different conventions than the ones we use. For relations betweenconventions used in several papers on knot contact homology, see [Ng14, page 527]).Recall the definition of the full linearized knot contact homology in Section 4.2. Lemma 4.25.
There is a Euclidean neighborhood U Ă ˜ V K of (cid:15) “ p , q such that, for every (cid:15) P U , { KCH (cid:15)k p K ; C q – $’&’% C if k “ C if k “ otherwise . Proof.
We begin with the computation for (cid:15) . As mentioned above, KCH (cid:15) ˚ p K ; C q was computedin [EN18, Section 4], see (4.10). Consider as above braiding K around the unknot U . Then K isrepresented as the closure of an n -strand braid. As shown in [EENS13], the linearized chain complex KCC (cid:15) ˚ p K ; C q has n p n ´ q generators in degree 0 (we denote them a i ), n p n ´ q generators indegree 1 (we denote them b j ) and n generators in degree 2.The vector space { KCC (cid:15) ˚ can be identified with KCC (cid:15) ˚ , with additional generators λ, µ, Q indegree 0. Note that p B (cid:15) “ B (cid:15) , so { KCH (cid:15) p K ; C q – KCH (cid:15) p K ; C q – C . We next show(4.14) dim p im p p B (cid:15) qq “ dim p im pB (cid:15) qq ` . Since
KCH (cid:15) p K ; C q “
0, this implies that dim p im p p B (cid:15) qq “ dim { KCC ´
2, hence { KCH (cid:15) p K ; C q “ C . The fact that { KCH (cid:15) p K ; C q – χ ´ { KCH (cid:15) ˚ p K ; C q ¯ “ χ p KCH (cid:15) ˚ p K ; C qq ` “ . We now prove (4.14). Observe that codim ` im pB (cid:15) q Ă ker pB (cid:15) q ˘ “ y p q is a non-zero vector in the kernel, in the complement of the image. Denote this vectorby β . Pick a basis for a complement of ker pB (cid:15) q Ă C b , and denote its members β j , 1 ď j ď n p n ´ q .Define new variables l “ λ ´ (cid:15) p λ q “ λ ´ m “ µ ´ (cid:15) p µ q “ µ ´ q “ Q ´ (cid:15) p Q q “ Q ´
1. Wecan then write the DGA differential of the β j as B β j “ ξ j p l, m, q q ` ÿ i ě ξ ji p l, m, q q a i ` O p a q where the ξ j and ξ ji are power series satisfying ξ j p , , q “ j and ξ i p , , q “ i .Our assumptions on the β j also imply that the square matrix ´ ξ ji p , , q ¯ ď j,i ď n p n ´ q is invertible. Now, p B (cid:15) p β q “ ˆ B ξ B l p , , q ˙ l ` ˆ B ξ B m p , , q ˙ m ` ˆ B ξ B q p , , q ˙ q. Corollary 4.21, implies that the q -derivative in the expression above is non-zero. The fact that p B (cid:15) p β j q “ ˜ B ξ j B l p , , q ¸ l ` ˜ B ξ j B m p , , q ¸ m ` ˜ B ξ j B q p , , q ¸ q ` B (cid:15) p β j q for all j ě p B (cid:15) p β q imply (4.14). To finish the proof we show thefollowing: If the lemma holds for some (cid:15) P ˜ V K , then it also holds for all other augmentations in a neigh-borhood of (cid:15) . This would follow from the upper semicontinuity of dim { KCH (cid:15) p K ; C q and of dim { KCH (cid:15) p K ; C q at an (cid:15) where the lemma holds. If these dimensions cannot increase near (cid:15) , then (using again thefact that the Euler characteristic of this complex is 3)dim { KCH (cid:15) p K ; C q “ dim { KCH (cid:15) p K ; C q ` dim { KCH (cid:15) p K ; C q ´ ď ` ´ “ { KCH (cid:15) p K ; C q “
0. The same Euler characteristic constraint then implies that thedimensions of { KCH (cid:15) p K ; C q and { KCH (cid:15) p K ; C q must also be constant near (cid:15) .If we proved that { KCH (cid:15) ˚ p K ; C q is a coherent sheaf on ˜ V K , then the upper semicontinuity of thedimensions of the cohomology stalks [Har77, Chapter III, Theorem 12.8] would give us the uppersemicontinuity of { KCH (cid:15)k p K ; C q for all k . We will however give a more pedestrian argument for k “ k “ c , we can think of p B (cid:15) p c q as the result of applying (cid:15) to all but one occurencesof λ, µ, Q or Reeb chords in each term in B c . This is clearly continuous in (cid:15) . By assumption, therank of p B (cid:15) is n ´
1. The continuity of p B (cid:15) guarantees that the rank of p B (cid:15) is also at least n ´ (cid:15) , which immediately implies the upper semicontinuity of dim { KCH (cid:15) p K ; C q at (cid:15) . Similarly, therank of p B (cid:15) is n p n ´ q ` (cid:15) , hence it is at least as much for nearby (cid:15) . This implies the uppersemicontinuity of dim { KCH (cid:15) p K ; C q . The lemma follows. (cid:3) Lemma 4.25 has the following consequence. We use notation as in its proof: a “ p a i q and b “ p b j q are degree zero and one Reeb chords of Λ K , respectively, β is a non-zero element in ker pB (cid:15) q inthe complement of im pB (cid:15) q , and β j P C b , j “ , . . . , n p n ´ q , is a basis of the complement ofker pB (cid:15) q . Let ˜ W K Ă p C ˚ q ˆ C a « p C ˚ q ˆ C n p n ´ q be the zero locus of the polynomials B β j , j “ , . . . , n p n ´ q . Corollary 4.26.
The solution set ˜ W K is a complex manifold of dimension 2 in a Euclidean neigh-borhood of p , q .Proof. The polynomials B β j define a map p C ˚ q ˆ C a Ñ C n p n ´ q` . The differential of this mapat p , q is given by p B (cid:15) , and was shown to have full rank. Indeed, the matrix of q “ Q ´ a partial derivatives is invertible. The implicit function theorem then implies that ˜ W K is smoothnear p , q , and locally parameterized by l “ λ ´ m “ µ ´ (cid:3) UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 25
Proof of Proposition 4.15 (1).
As we pointed out earlier, the statement for (cid:15) “ p , q followsfrom [EN18, Section 4]. The upper semicontinuity of the ranks of cohomology groups implies that,for (cid:15) near (cid:15) , KCH (cid:15) ˚ p K ; C q is either as stated or it vanishes. The argument at the end of the proofof Lemma 4.25 can be adapted to show this semicontinuity. Let us now assume that KCH (cid:15) ˚ p K ; C q vanishes for (cid:15) arbitrarily close to (cid:15) . Since p B (cid:15) “ B (cid:15) , we get { KCH (cid:15) p K ; C q “
0, which contradictsLemma 4.25. (cid:3)
Proof of Proposition 4.11.
It is sufficient to prove the statement for a Euclidean neighborhoodof p , q in ˜ V K , since the singular locus of an affine variety is Zariski-closed. Corollary 4.26 impliesthat the vanishing locus of the polynomials B β j is a smooth complex variety ˜ W K of dimension 2near p , q . We will show that ˜ V K coincides with ˜ W K near p , q , which is to say that the B ˜ β k vanish in a neighborhood of p , q in ˜ W K .Pick a collection t c i u ď i ď n ´ of Reeb chords of degree 2, such that the ˜ β i – B (cid:15) p c i q are linearlyindependent. These exist, by Proposition 4.15 (1). Note that t β j , ˜ β i u , with 0 ď j ď n p n ´ q and1 ď i ď n ´
1, is a basis of C b “ KCC (cid:15) p K ; C q for any (cid:15) .For all k , we have B c k “ α k p λ, µ, Q q ˜ β k ` ÿ i O p a q ˜ β i ` ÿ j O p a q β j , where the coefficients O p a q depend on p λ, µ, Q, a q and have order at least one in a , and α k p , , q “
1. Since B “
0, we get(4.15) α k B ˜ β k “ ÿ i O p a qB ˜ β i ` ÿ j O p a qB β j . Assume that, for some k , B ˜ β k does not vanish near p , q in ˜ W K . This will lead us to a con-tradiction. We know that ˜ W K is 2-dimensional and parametrized by λ ´ µ ´ p , q .The restriction of B ˜ β k to ˜ W K can be written as a power series in λ ´ µ ´ p , q . Byassumption, this series is non-zero, and has a minimal order in λ ´ µ ´
1. Pick the k for whichthis minimal order is the smallest, and denote this order by s . The variables a can also be expressedas power series in λ ´ µ ´ p , q in ˜ W K ), and we know that the constant terms in theseseries vanish. Now, restricting equation (4.15) to ˜ W K , and using the fact that α k p , , q ‰ B β j vanish along ˜ W K , we conclude that the right side of the equation has order greaterthan s , which is a contradiction. (cid:3) A 2-parameter family of generators of linearized homology.
By Proposition 4.11, we can pa-rametrize ˜ V K by coordinates λ, µ near p , q . Let (cid:15) λ,µ denote the augmentation at p λ, µ q . Consideran analytic family y p λ, µ q of cycles in KCC (cid:15) λ,µ p K ; C q generating KCH (cid:15) λ,µ p K ; C q (which, possiblyafter restricting to a smaller open set, is 1-dimensional by Proposition 4.15(1)). To see why suchan analytic family exists we argue as for y p λ q in Definition 4.17: we have analytic maps U Ñ Fl n p k, k ` qp λ, µ q ÞÑ ` im B (cid:15) λ,µ , ker B (cid:15) λ,µ ˘ where U is an open subset of C and Fl n p k, k ` q is the partial flag variety of k and k ` C n . We can thus think of ker B (cid:15) λ,µ as defining a holomorphic vector bundle E over U , with a holomorphic subbundle E given by im B (cid:15) λ,µ . The analytic function y p λ, µ q can bedefined locally by taking a section of E that is transverse to E . After multiplication of y p λ, µ q byan analytic function, we can assume that y p λ, q “ y p λ q , where y p λ q is as in Definition 4.17. In amanner similar to that definition, we have the following. Definition 4.27.
Let F p λ, µ, Q q – Bp y p λ, µ qq| a ÞÑ (cid:15) λ,µ p a q , where B is the DGA-differential on A K . Proof of Proposition 4.15 (2).
We show that(4.16) B F B Q p λ, , Q q “ B G B Q p λ, , Q q and(4.17) B F B µ p λ, , q “ B G B µ p λ, , q . Then (4.16) and Corollary 4.21 imply (4.6), and (4.17), together with Corollary 4.19, implies (4.7).To see that (4.16) and (4.17) hold, consider the following Taylor expansions in µ ´ y p λ, µ q “ y p λ q ` p µ ´ q z p λ q ` O pp µ ´ q q (cid:15) λ,µ “ (cid:15) λ ` p µ ´ q η λ ` O pp µ ´ q q for suitable holomorphic functions z and η . We can then write, F p λ, µ, Q q “ G p λ, µ, Q q ` p µ ´ q ` pBp y p λ qqq| a ÞÑp (cid:15) λ ,η λ qp a q ` pBp z p λ qqq| a ÞÑ (cid:15) λ p a q ˘ ` O pp µ ´ q q where x a ÞÑp (cid:15) λ ,η λ qp a q is the sum of all possible ways of applying (cid:15) λ to all but one of the a i in x , andapplying η λ to that chosen a i in x . Equation (4.16) follows immediately. On the other hand, B F B µ p λ, , q “ B G B µ p λ, , q ` ` pBp y p λ qqq| a ÞÑp (cid:15) λ ,η λ qp a q ` pBp z p λ qqq| a ÞÑ (cid:15) λ p a q ˘ | µ “ Q “ ““ B G B µ p λ, , q ` η λ ` B (cid:15) λ p y p λ qq ˘ ` (cid:15) λ pBp z p λ qqq . The claim now follows from the fact that the two last summands vanish. The first is zero because y p λ q is a cycle in the chain complex linearized by (cid:15) λ . The second vanishes since (cid:15) λ ˝ B “ (cid:3) From knot contact homology to the Alexander polynomial
In this section we prove the main result of this paper, Theorem 1.2. We first discuss the gluingtheorems associated to degenerations of annuli that do not have counterparts for disks. We thenprove the theorem by a series of identifications of boundaries of moduli spaces.5.1.
Gluing results for moduli spaces of once punctured annuli.
The key point in ourargument below concerns equating quantities that count ends of one-dimensional moduli spaces.To control compactifications of moduli spaces there are two ingredients: compactness and gluing.Compactness controls the possible limits of a sequence holomorphic curves. Here we use Gromovcompactness and SFT compactness. Gluing produces a neighborhood in the moduli space of abroken configuration in such a limit. Here we use Floer gluing, which can be described as aninfinite dimensional version of Newton iteration. We consider the results on disks as standard andfocus on two gluing theorems that arise for annuli.The first case of degeneration and gluing for annuli was called elliptic boundary splitting in[ES19], and is depicted in Figure 4. It corresponds to the limit where the modulus of the annulusconverges to infinity, by having one of its boundary loops shrink to a point, and can be described asfollows. Use coordinates p z , z , z q “ p x ` iy , x ` iy , x ` iy q on C . The Lagrangian boundarycondition corresponds locally to R “ t y “ y “ y “ u . Take the vector field s “ B y , and the4-chain V “ tp z , z , z q : y “ y “ , y ě u , oriented so that it induces the positive orientation on R .Consider the family of maps u ρ : r´ ρ,
8q ˆ S Ñ C , given by(5.1) u ρ p ζ q “ ´ e ζ ` e ´ ζ ´ ρ , ´ i p e ζ ´ e ´ ζ ´ ρ q , ¯ , UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 27 R R Ñ Ð Vu t v t Figure 4.
Elliptic boundary splittingwhere the domain is thought of as a subset of the cylinder C { πi Z . Note that u ρ pt´ ρ u ˆ S q isa circle of radius 2 e ´ ρ in the x x -plane and that the interior of u ρ is disjoint from the 4-chain.Consider also the family v t : r´8 ,
8q ˆ S {pt´8u ˆ S q Ñ C given by v t p ζ q “ p e ζ , ´ ie ζ , it q , t P p´ (cid:15), (cid:15) q , where we think of the domain as a smooth disk. The intersection v t X V is the point p , , it q . As ρ Ñ 8 the map u ρ converges to v , which corresponds to when the family v t crosses the Lagrangian R . This allows us to identify the nodal curve u with the smooth curve v . We again write u t forthe reparameterized family with t “ { ρ .Boundaries of a generic 1-parameter family of annuli in which the conformal ratio is going toinfinity are locally modelled on the above family, in the following sense. Let a t for t P r , (cid:15) q and b t for t P p´ (cid:15), (cid:15) q be two families of maps to p T ˚ R , L q , where L is a Lagrangian (in practice L “ R ,below), such that a is a map from a domain with an elliptic node, with the node mapping to p P L ,and b is the corresponding map from the normalization of a .Then the pair p a t , b t q is a standard elliptic degeneration if there exists a neighborhood U around p that can be identified with a neighborhood of the origin in C with the Lagrangian corresponding to R and an explicit 4-chain such that there is a diffeomorphism U Ñ C that respects the Lagrangianand the 4-chain and carries the intersections of the curves in the family with U to the curves in themodel family p u t , v t q above. If instead a t is a nodal family for t P p´ (cid:15), s , we again say the pair isa standard elliptic degeneration if there is an identification as above with time reversed.Let L Ă T ˚ R be a Lagrangian submanifold asymptotic to a Legendrian Λ Ă ST ˚ R and let c be a degree one Reeb chord of Λ. Write A ˝ and D ˝ for an annulus and a disk, respectively, with aboundary puncture. Lemma 5.1.
Let u t : p A ˝ t , B A ˝ t q Ñ p T ˚ R , R Y L q be a transversely cut out 1-parameter family ofannuli with a positive puncture at c , and let v t : p D ˝ , B D ˝ q Ñ p T ˚ R , L q be a 1-parameter family ofdisks with a positive puncture at c and intersecting V transversely at an interior marked point. If u “ v , then the pair p u t , v t q is a standard elliptic degeneration.Proof. This is [ES19, Lemma 4.16]. (cid:3)
The second degeneration concerns instants when annuli without positive punctures split off of 1-parameter families of annuli with one positive puncture, it was called hyperbolic boundary splittingin [ES19], see Figure 5. The limit configuration consists of a rigid annulus and a disk intersectingit at a boundary point. The annulus may be multiply covered, however it inherits a boundarymarked point from the punctured disk that breaks the rotational symmetry of the multiple cover.With these marked points the gluing becomes standard: one disk family is glued to one annuluswith boundary marked point.More specifically, let M ˚ p c q denote the moduli space of disks with one boundary puncture at 1mapping to a Reeb hord c of index 1 and a boundary marked point at ´
1. The virtual dimensionof this space is 2. Let M ˚ an p L q denote the moduli space of annuli with a boundary marked point at1 (in the boundary component mapping to L ). Its virtual dimension is 1. We observed in Section3 that, in our setting, we can assume both moduli spaces to be transversely cut out. The moduli c L R Õ c R L Ñ L c R L Ð Lc R L L
Figure 5.
Horizontally, a family of disks crossing a rigid annulus. Diagonally,hyperbolic boundary splitting of a family of annuli with a boundary puncturespace of split annuli is p ev ˆ ev ´ q ´ p ∆ L q , whereev ˆ ev ´ : M ˚ an p L q ˆ M ˚ p c q Ñ L ˆ L is the product of evaluation maps and ∆ L Ă L ˆ L is the diagonal. Adapting arguments frome.g. [ES19, Section 4.4.1] or [MS04], one shows that ev ˆ ev ´ can be assumed transverse to ∆ L .Also, p ev ˆ ev ´ q ´ p ∆ L q is a boundary stratum of the Gromov compactification of the modulispace M ˚ an p c, L q of annuli with one boundary puncture at c . Lemma 5.2.
Let v t : p D ˝ , B D ˝ q Ñ p T ˚ R , L q be a transversely cut out 1-parameter family of diskswith one boundary puncture at c and let u : p A r , B A r q Ñ p T ˚ R , R Y L q be an embedded rigidannulus, such that v t intersects the boundary of u p A r q transversely at t “ . Then, for each suchintersection point and each d ą , there is a 1-parameter family of annuli with one positive punctureat c and whose Gromov-limit consists of v and the d -fold cover of u .In other words, there is a neighborhood of p ev ˆ ev ´ q ´ p ∆ L q in the Gromov compactificationof M ˚ an p c, L q that is diffeomorphic to the product of p ev ˆ ev ´ q ´ p ∆ L q with the interval r , q .Proof. Since the markers kill the automorphisms of the annuli, the proof reduces to the well-known case of gluing of holomorphic curves with two boundary components meeting with transverseevaluation maps at a nodal point, see for example [FOOO09, BC07]. (cid:3)
Remark . Lemma 5.2 is a simple example of a gluing theorem where multiple covers are leftunperturbed. In the present situation, only one of the components is multiply covered and since itis an annulus only regular covers contribute. In general, when two simple curves that both admitformally rigid multiple covers intersect on the boundary, one must take into account contributionsfrom all of the branched covers with constant curves attached when gluing. For simple disks, suchformulas were worked out from a physics perspective in [EKL20], where it is also shown that the caseof simple curves of higher genus [EKL20, Appendix A] is more involved, with corrections to the ‘two-to-three-formula’, [EKL20, Equation (6.24)]. It would be valuable to have a more mathematicalperspective on these gluing formulas, e.g. from the perspective of obstruction bundles.5.2.
The boundary of the space of punctured annuli.
Let K Ă R be a knot and let Λ K beits Legendrian knot conormal. We write L for a Lagrangian filling of Λ K , where either L “ L δK or L “ M δK . We assume that L X R is transverse and consists of a collections of points ξ j of index 1and η j of index 2. (For L δK , there are no points in the intersection and for M δK , intersection pointscorrespond to critical points of a circle valued Morse function.) UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 29
Recall that we have an analytic family of cycles y p λ, µ q in KCC (cid:15) λ,µ that generates the linearizedhomology. Here we will write y “ y p λ, q if L “ L δK and y “ y p , µ q if L “ M δK .Define the moduli space M an p y q to be the following weighted 1-manifold. Let y “ ř m j b j , where m j are functions of λ or µ when L “ L δK and L “ M δK , respectively. Let M an p y q “ ÿ j m j M an p b j q , where M an p b j q is the moduli space of holomorphic annuli with one positive boundary puncture at b j , one boundary component on R and the other on L . See Figure 6 for a depiction of this modulispace and its boundary components, which we now discuss.We use notation as above: when we say that curves in a moduli space have positive puncture at y this means that we take the m j -weighted sum of the corresponding moduli spaces with positivepuncture at c j . The moduli spaces are as follows: ‚ The fibered product over boundary evaluation maps of moduli spaces M p y qˆ ev B M an p R , L q ,where M p y q is the moduli space of holomorphic disks with boundary on L and one positivepuncture at y and M an p R , L q the moduli space of annuli with one boundary componenton R and the other on L . ‚ The moduli space M p y ; L q of holomorphic disks in T ˚ R , with boundary on L and aninterior marked point mapping to R . ‚ The union of products of moduli spaces M an p y, a q : “ Ť j M p y, a j q ˆ M an p a j q and where M an p a j q , where M p y, a j q is the moduli space of strips with positive puncture at y andnegative puncture at a j , as in the linearized differential, is the moduli space of annuli withpositive boundary puncture at a j , one boundary component on L and the other on R . ‚ The fibered product over boundary evaluation maps of moduli spaces M an p y q ˆ ev B M p L q ,where M p L q is the moduli space of holomorphic disks with boundary on L . ‚ The union of moduli spaces M p y, ξ q “ Ť j M p y, ξ j , ξ j q , where M p y, ξ j , ξ j q is the space ofholomorphic disks with boundary on L Y R with three boundary punctures, one positivepuncture at y and two punctures at the intersection point ξ j . ‚ The union of moduli spaces M p y, η q “ Ť j M p y, η j , η j q , where M p y, η j , η j q is the space ofholomorphic disks with boundary on L Y R with three boundary punctures, one positivepuncture at y and two punctures at the intersection point η j . Lemma 5.4.
For a generic almost complex structure, the moduli space M an p y q is transversely cutout and forms a weighted 1-manifold with a natural compactification. Its boundary is B M an p y q “ M p y q ˆ ev B M an p R , L q Y M p y ; L q Y M an p y, a q Y M an p y q ˆ ev B M p L qY M p y, ξ q Y M p y, η q . (5.2) Here, M an p y, a q has algebraically zero points since y is a cycle and M an p y q ˆ ev B M p L q has alge-braically zero points since M p L q is empty by Lemma 3.2.Proof. Note that every curve in M p y q has injective points near the positive puncture, and hencestandard arguments show that perturbing the almost complex structure suffices to achieve transver-sality. To understand the boundary, we consider degenerations of the domain where the limit iseither a nodal curve with one disk and one annulus component or the conformal ratio of the annulusgoes to infinity. The nodal limits are curves in M p y q ˆ ev B M an p R , L q Y M an p y, a q Y M an p y q ˆ ev B M p L q . To parameterize a neighborhood of the boundary of M an p y q , we use Lemma 5.2 for M p y q ˆ ev B M an p R , L q , for M an p y, a q we use SFT-gluing, see e.g. [CELN17, Section 10] and for M an p y q ˆ ev B M p L q we use standard boundary gluing, see [ES16, Section 6]. R ˆ Λ K y y y y yaL R R R L L L R R L B “ Y Y Y M an p y q M p y q ˆ ev B M an p R , L q M p y ; L q y L R Y ξ i y L R η j Y M an p y, a q M an p y q ˆ ev B M p L q M p y, ξ q M p y, η q Figure 6.
The boundary of M an p y q Degeneration of the conformal ratio of the annulus corresponds to M p y ; L q Y M p y, ξ q Y M p y, η q , where a neighborhood of the boundary is parameterized via Lemma 5.1 for M p y, L q (which corre-sponds to when the modulus of the annulus goes to infinity), and gluing at a Lagrangian intersectionpoint for M p y, ξ qY M p y, η q (corresponding to when the modulus goes to zero), see [ES14, AppendixA.4–6]. (cid:3) Curve counts at infinity.
In this section, we find 1-dimensional moduli spaces which serve ascobordisms between the boundary components of M an p y q in (5.2) that do not involve Lagrangiandouble points and holomorphic curves at infinity, namely M p y q ˆ ev B M an p R , L q , M p y ; L q and M an p y q ˆ ev B M p L q . In order to treat L “ L δK and L “ M δK simultaneously, we consider curves α and β in Λ K that generate H p Λ K , Z q , so that α generates H p L, Z q and β is null-homologous in L . (I.e., α “ x and β “ p if L “ L δK and vice versa if L “ M δK .)We start with M p y qˆ ev B M an p R , L q . The construction here is directly analogous to the construc-tion for the disk potential in [AENV14]. Each holomorphic annulus u : A R Ñ T ˚ R in M a n p R , L q ,defines, by restricting to the boundary component B A R that maps to L , an element in H p L ; Z q .An area argument shows that this is a positive multiple k of the generator α . Fix a smooth Borel–Moore 2-chain σ u in L giving a homology between the loop u pB A R q and kα in Λ K at infinity. Wecall σ u a bounding chain , as in [AENV14, Section 6.3].Let σ “ ď u P M an p L q σ u denote the union of all bounding chains. Recall that there are finitely many annuli below anygiven action. Define M p y, σ q as the 1-dimensional moduli space of holomorphic disks with positivepuncture at y and a boundary point mapping to σ . See Figure 7.We next describe the boundary of M p y, σ q in terms of other moduli spaces as follows. UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 31 R ˆ Λ K y R L B “ Y y R L y R L L δ K M p y q ˆ ev B M an p R , L q M p y, σ q M p y, σ q σ u σ u u u R L Lσ u u ya Y u R ˆ Λ K u u u u M p y, a , σ q Figure 7.
The boundary of M p y, σ q Ly a R ˆ Λ K y L ya La R ˆ Λ K R ˆ Λ K Y Y Y ¨ ¨ ¨
Figure 8.
The components at infinity in M p y, σ q‚ Let M p y, σ q denote the curves in M p y, σ q at infinity, i.e., the curves associated to R -invariant curves in Λ K ˆ R in M p y, σ q . The schematic representation of M p y, σ q inFigure 7 should be thought of as a simplification. A more accurate illustration of the termsat infinity is given in Figure 8. Similar observations should be made about curves at infinityin the various moduli spaces considered in this section. ‚ Let M p y, a , σ q denote the union of products M p y, a , σ q “ ď j M p y, a j q ˆ M p a j , σ q . Lemma 5.5.
The boundary of the moduli space M p y, σ q is M p y, σ q Y M p y q ˆ ev B M an p R , L q Y M p y, a , σ q , where M p y, a , σ q contains algebraically zero points since y is a cycle.Proof. There are three sources of non compactness: either the curve breaks which corresponds to M p y, a , σ q , or it moves to infinity without breaking which corresponds to M p y, σ q , or the markedpoint moves to the boundary of σ which corresponds to M p y q ˆ ev B M an p R , L q , by definition of σ . (cid:3) We next consider the moduli space M p y, L q in a similar spirit. Fix a smooth Borel–Moore 4-chain V in T ˚ R whose boundary is R . Explicitly, pick a section s of the trivial bundle ST ˚ R Ñ R (as in Section 4.1) and define(5.3) V : “ t ts p x q | t ě , x P R u Ă T ˚ R . τ Ă Λ K V X L “ γσ τ Ă L τ
Figure 9.
A component of V X L “ γ , with τ Ă Λ K such that B σ τ “ γ ` τ B “ Y y R LyV L M p y, V q M p y, L q R ˆ Λ K y ya B V L Y y γL Y V M p y, γ q M p y, B V q M p y, a , V q Figure 10.
The boundary of M p y, V q We then identify the boundary at infinity of V , which we denote by B V , with the section s .We also need to take into account the fact that V X L ‰ ∅ . For generic s , V X L is an oriented1-manifold γ with boundary at infinity B γ “ B V X Λ K , which is an oriented 0-manifold. Wepick a 1-chain τ Ă Λ K such that B τ “ B γ . We take this 1-chain so that the 1-cycle γ ` τ isnull-homologous in L . Finally, pick a bounding 2-chain σ τ for γ ` τ . See Figure 9. Remark . The choice of τ is not canonical. More precisely, τ is defined up to a multiple k ¨ β ofthe curve β in Λ K , which is null-homologous in L .We now introduce relevant moduli spaces. Define M p y, V q as the moduli space of holomorphic disks with boundary on L , with one positive boundary punctureat y and one interior marked point mapping to V , see Figure 10. This is a 1-dimensional modulispace. Also, let M p y, σ τ q denote the moduli space of holomorphic disks with boundary on L , with one positive boundarypuncture at y and one boundary marked point mapping to σ τ , see Figure 11. This is a 1-dimensionalmoduli space as well.Relevant 0-dimensional moduli spaces are the following: ‚ The moduli space M p y, B V q of holomorphic disks at infinity (divided by the R -translationin the symplectization), with a positive boundary puncture at y and an interior marked pointmapping to B V . ‚ The moduli space M p y, γ q of holomorphic disks with a positive boundary puncture at y and a boundary marked point mapping to γ . UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 33 R ˆ Λ K B “ Y y σ τ L y γL M p y, σ τ q M p y, γ q y τ M p y, τ q ya L Y σ τ M p y, a , σ τ q Figure 11.
The boundary of M p y, σ τ q‚ The moduli space M p y, a , V q : the union of products of moduli spaces M p y, a j qˆ M p a j , V q ,where M p a j , V q is the moduli space of disks with positive puncture at a j boundary on L and an interior marked point mapping to V . ‚ The moduli space M p y, τ q of holomorphic disks at infinity, with a positive boundarypuncture at y and a boundary marked point mapping to τ (divided by the R -translation inthe symplectization). ‚ The moduli space M p y, a , σ τ q : the union of products of moduli spaces M p y, a j qˆ M p a j , σ τ q ,where M p a j , σ τ q is the moduli space of disks with positive puncture at a j boundary on L and a boundary marked point mapping to σ τ . Lemma 5.7.
The 1-dimensional moduli space M p y, V q has a natural compactification with bound-ary M p y, L q Y M p y, B V q Y M p y, γ q Y M p y, a , V q , where M p y, a , V q contains algebraically zero points since y is a cycle.Proof. The boundary of M p y, V q is given by three degenerations. First, the interior marked pointcan move to the boundary of V , which gives M p y, L q . Second, the holomorphic curves can move tothe boundary in the moduli space of maps. This corresponds here to the curve moving to infinitywhich gives M p y, B V q , or to SFT-breaking which gives M p y, a , V q . Third, the interior markedpoint can move to the boundary of the domain, which corresponds to M p y, γ q . (cid:3) Similarly, we have the following result.
Lemma 5.8.
The 1-dimensional moduli space M p y, σ τ q has a natural compactification with bound-ary M p y, γ q Y M p y, τ q Y M p y, a , σ τ q , where M p y, a , σ τ q contains algebraically zero points since y is a cycle.Proof. The boundary of M p y, τ q is given by two degenerations. First, the marked point can moveto the boundary of σ τ , which gives M p y, γ q . Second, the holomorphic curves can move to theboundary in the moduli space of maps. This corresponds here to the curve moving to infinitywhich gives M p y, τ q , or to SFT-breaking which gives M p y, a , σ τ q . (cid:3) We next apply these identifications of boundaries of moduli spaces to derive formulas for actualcurve counts for L “ L δK . Recall the function G p λ, µ, Q q “ pB y p λ qq| a ÞÑ (cid:15) LK,λ p a q that was introducedin Definition 4.17, and the power series An L K p λ q from Section 3.1. Recall the identification ofvariables for L “ L δK : α “ x and β “ p . Recall also that λ “ e x , µ “ e p , and Q “ e t . Then wehave the following result. R ˆ Λ K y R L δ K M p y, σ q σ u R ˆ Λ K Y R ˆ Λ K y B V M p y, B V q y τ M p y, τ qY Figure 12.
A union of moduli spaces which is a boundary.
Proposition 5.9.
The following equation holds for any knot K : (5.4) pB p G p e x , , qq ¨ pB x An L K p e x qq ` pB t G p e x , , qq “ . Proof.
We use Lemma 5.4. Since L δK X R “ ∅ the last two moduli spaces in equation (5.2) areempty. Writing | M | for the algebraic count of points in the weighted 0-dimensional space M , weget the equation: | M p y q ˆ ev B M an p R , L q| ` | M p y ; L q| “ . Lemma 5.5 gives | M p y, σ q| ´ | M p y q ˆ ev B M an p R , L q| “ , Lemma 5.7 gives | M p y ; L q| ´ | M p y, B V q| ` | M p y, γ q| “ | M p y, γ q| ` | M p y, τ q| “ . We conclude that(5.5) | M p y, σ q| ` | M p y, B V q| ` | M p y, τ q| “ , see Figure 12. Now, | M p y, σ q| counts the number of insertions of σ (the bounding chain of annuli)in disks at infinity with one positive puncture on y and boundary in R ˆ Λ K , and this is(5.6) | M p y, σ q| “ pB p G p e x , , qq ¨ ´ B x An L δK p e x q ¯ . This uses the calculation of the λ - and µ -powers in A K explained in Section 4.1. It is also usefulto observe that the coefficient of an annulus u in B x An L δK p e x q is (up to sign) given by d p u q{ m p u q (degree over multiplicity). This is an integer number, specifically the intersection number of theboundary of u with a surface Poincar´e-dual to the longitude curve x .Also, | M p y, B V q| ` | M p y, τ q| counts disks at infinity with an interior puncture in B V or aboundary puncture in τ . By Lemma 4.2, there are choices of capping surfaces Σ x , Σ p such that(5.7) | M p y, B V q| ` | M p y, τ q| “ B t G p e x , , q . Equations (5.5) – (5.7) give the result. (cid:3)
UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 35 y M δ K R η j ξ i y M δ K R η j ξ i M δ K M δ K B “ y M δ K R η j η k M δ K ξ i ξ k Y y M δ K R η j ξ i M δ K Y M p y, ξ i , η j q Ť k M p ξ i , η k q ˆ M p y, η k , η j q Ť k M p y, ξ i , ξ k q ˆ M p ξ k , η j q M p y q ˆ ev B M p ξ i , η j q a M δ K R η j ξ i M δ K M p y, a , ξ i , η j q y Y Figure 13.
The boundary of M p y, ξ i , η j q Remark . Recall the non-uniqueness of τ , see Remark 5.6. Picking a different τ differing fromthe one above by k ¨ p would change the equation (5.4) to pB p G p e x , , qq ¨ pB x An L K p e x qq ` pB t G p e x , , qq ` k pB p G p e x , , qq “ , corresponding to a change of variables t ÞÑ t ` kp .5.3.1. Proof of Proposition 4.20.
According to Corollary 3.7,An L K p e x q “ ÿ k ą e kx k “ ´ log p ´ e x q , so pB x An L K p e x qq “ e x ´ e x “ λ ´ λ . Combined with Proposition 5.9, this implies Proposition 4.20.5.4.
Counting Floer strips.
In the previous section we expressed the first two non-zero boundaryterms in Lemma 5.4, see (5.2), in terms of curve counts at infinity. Here we will carry out a similaranalysis of the last two terms.We now take L “ M δK and as above we denote by ξ “ t ξ i u the intersection points in L X R ofindex 1 and by η “ t η i u those of index 2. We consider the following moduli spaces, illustrated inFigure 13. ‚ The 1-dimensional moduli space M p y, ξ i , η j q of holomorphic disks with a positive boundarypuncture at y and two Lagrangian intersection punctures at intersection points ξ i and η j . ‚ The 0-dimensional moduli space M p y, ξ i , ξ j q of holomorphic disks with a positive boundarypuncture at y and two Lagrangian intersection punctures at intersection points ξ i and ξ j . ‚ The 0-dimensional moduli space M p y, η i , η j q of holomorphic disks with a positive boundarypuncture at y and two Lagrangian intersection punctures at intersection points η i and η j . ‚ The union of products of moduli spaces M p y, a , ξ i , η j q “ ď a k P a M p y, a j q ˆ M p a k , ξ i , η j q . ‚ The 0-dimensional moduli space M p ξ i , η j q of holomorphic strips with one puncture at ξ j and the other at η j .Recall that M p y q is the moduli space of of holomorphic disks with boundary on M δ K and onepositive boundary puncture asymptotic to y . B M δ K R η j ξ i M δ K Y y M δ K R η j ξ i ˜ σ u u y M δ K R η j ξ i ˜ σ u u R ˆ Λ K “ M δ K y M p y, ξ i , η j , ˜ σ q M p y, ξ i , η j , ˜ σ q M p y q ˆ ev B M p ξ i , η j q Y a M δ K R η j ξ i ˜ σ u u M δ K M p y, a , ξ i , η j , ˜ σ q yu M δ K R η j ξ i M δ K Y y M p y, γ ij q u γ ij Figure 14.
The boundary of M p y, ξ i , η j , ˜ σ q Lemma 5.11.
The boundary of the 1-dimensional moduli space M p y, ξ i , η j q consists of the modulispaces ď k M p ξ i , η k qˆ M p y, η k , η j q Y ď k M p y, ξ i , ξ k qˆ M p ξ k , η j q Y M p y qˆ ev B M p ξ i , η j q Y M p y, a , ξ i , η j q , where M p y, a , ξ i , η j q has algebraically zero points since y is a cycle. Here, ev B are evaluation mapsat boundary components mapping to M δ K .Proof. The boundary correspond to three types of breaking. Breaking at an intersection point gives Ť k M p ξ i , η k q ˆ M p y, η k , η j q and Ť k M p y, ξ i , ξ k q ˆ M p ξ k , η j q . Boundary breaking gives M p y q ˆ ev B M p ξ i , η j q . Finally, breaking at a Reeb chord gives M p y, a , ξ i , η j q . (cid:3) We next pick reference paths γ ij connecting the intersection points ξ i and η j in M δ K . This willbe done by fixing a base point, picking paths γ ξ i , γ η j connecting to the base point, and taking γ ij “ γ ξ i Y´ γ η j . Pick also bounding chains ˜ σ u in M δ K for the boundaries of u P M p ξ i , η j q in M δ K , completedby the reference paths. We write M p y, ξ i , η j , ˜ σ q for the moduli space of disks with a positiveboundary puncture at y , and a boundary marked point mapping to ˜ σ u , where u is a holomorphicstrip between ξ i and η j , see Figure 14. Also, let M p y, ξ i , η j , ˜ σ q denote the moduli space of disksat infinity with a positive boundary puncture at y and a boundary marked point mapping to ˜ σ u ,and M p y, γ ij q the moduli space of disks with a positive boundary puncture at y and a boundarymarked point mapping to γ ij . Finally, define M p y, a , ξ i , η j , ˜ σ q “ Ť k M p y, a k q ˆ M p a k , ξ i , η j , ˜ σ q . Lemma 5.12.
The boundary of the moduli space M p y, ξ i , η j , ˜ σ q is M p y, ξ i , η j , ˜ σ q Y M p y q ˆ ev B M p ξ i , η j q Y M p y, γ ij q Y M p y, a , ξ i , η j , ˜ σ q , where M p y, a , ξ i , η j , ˜ σ q has algebraically zero points since y is a cycle, and where we can choose γ ij so that M p y, γ ij q has algebraically zero points.Proof. The space M p y, ξ i , η j , ˜ σ q corresponds to curves moving to infinity and M p y, a , ξ i , η j , ˜ σ q corresponds to SFT-breaking. The remaining terms correspond to the boundary marked pointmoving to the boundary of ˜ σ . To see that the last statement is true, note that B x F p , µ, q ‰ γ ij a suitable multiple of the curve p nearinfinity. (cid:3) Pick reference paths γ ij so that the last statement in Lemma 5.12 holds. Define the matrices E ξ , E η , and B (5.8) p E ξ q ij “ | M p ξ i , ξ j q| , p E η q ij “ | M p η i , η j q| , D ij “ | M p ξ i , η j q| . UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 37 c M δ K R ξ j E η p i, j q ξ i c M δ K R η j η i M δ K M δ K E ξ p i, j q Figure 15.
The terms counted in the entries of the matrices E ξ and E η Proposition 5.13.
Taking λ “ e x , µ “ e p and Q “ e t , we get (5.9) pB x F p , e p , qq “ B p An M K p e p q ` Tr ` D ´ pB p D qp e p q ˘‰ ` pB t F p , e p , qq “ , where Tr denotes the trace of a matrix. It follows that (5.10) ∆ K p e p q “ p ´ e p q exp ˆż ´ ˆ B t F B x F ˙ ˇˇˇ p ,e p , q dp ˙ . Proof.
Taking into account the contributions from the last two terms in Lemma 5.4 and repeatingthe argument in the proof of Proposition 5.9 gives(5.11) pB x F p , e p , qq ¨ pB p An M K p e p qq ` pB t F p , e p , qq ` Tr E ξ ` Tr E η “ . On the other hand, Lemmas 5.11 and 5.12 give E ξ ¨ D ` D ¨ E η “ | M p y, ξ i , η j , ˜ σ q| “ pB x F p , e p , qq ¨ pB p D q . The last identity uses again the description of the λ - and µ -powers in A K in Section 4.1. Multiplyingby D ´ and taking the trace, we get(5.12) Tr E ξ ` Tr E η “ pB x F p , e p , qq ¨ Tr ` D ´ pB p D q ˘ . Equation (5.9) now follows from combining equations (5.11) and (5.12).To get (5.10) from (5.9), first note that Proposition 4.7(2) implies that the denominator in theintegrand is not identically zero. Then, observe that p ´ e p q exp ˆż ´ ˆ B t F B x F ˙ ˇˇˇ p ,e p , q dp ˙ ““ p ´ e p q exp ˆż B p An M K p e p q dp ` ż Tr ` D ´ pB p D q ˘ dp ˙ ““ p ´ e p q exp p An M K p e p qq ¨ exp ˆż B p p log det D p e p qq dp ˙ ““ p ´ e p q exp p An M K p e p qq ¨ det D p e p q ““ ∆ K p e p q . In the last identity, we used Theorem 1.3. (cid:3)
Remark . If K is fibered, then M δ K can be made to not intersect the zero section in T ˚ R , asexplained in [AENV14]. In that case, the proof of Proposition 5.13 would be simpler, since onewould not need to take into account Floer strips. Remark . In the proof of Proposition 5.13, we used Theorem 1.3, which is a holomorphic curvereformulation of equation (2.3) where flow loops get replaced with annuli and flow lines are replacedwith strips. One should be careful that the choices made to specify the µ -powers for flow lines in τ Morse and for strips in τ str are comptible. The latter depend on the choices of paths γ ξ i , γ η j to thebasepoint. Observe that changes in those paths would translate into multiplying the matrix D p µ q in (5.8) on the left and on the right by diagonal matrices whose diagonal entries are powers of µ .The affect of that in Theorem 1.3 is to multiply τ str “ det p D p µ qq by some power of µ , which iscompatible with the fact that ∆ K p µ q is defined up to multiplication by such powers.5.5. Proof of Theorem 1.2.
Equation (1.1) in Theorem 1.2 is (5.10) in Proposition 5.13. To prove(1.2), we need to show that we can replace F with Aug K in Proposition 5.13, if B λ Aug K | p λ,Q q“p , q ‰
0. Recall that, according to Proposition 4.11, ˜ V K can be parametrized by p λ, µ q near p , q . Wedenote this parametrization (cid:15) λ,µ as before. Write Q p λ, µ q for the Q -component of (cid:15) λ,µ and considerthe graph of Q p λ, µ q : V – tp λ, µ, Q p λ, µ qqu Ă p C ˚ q . Observe that V is the image under π of ˜ V K near p , q . Note that F p λ, µ, Q p λ, µ qq “ (cid:15) λ,µ pBp y p λ, µ qqq “ (cid:15) λ,µ is an augmentation), so F | V ”
0. Also, equation (4.6) implies that p ∇ F q| ‰
0, so V is locally cut out by F near “ p , , q .We next show that V Ă V K . If Aug K were the zero polynomial for some knot K (which isconjecturally never the case, as recalled in Remark 4.14), then V K “ p C ˚ q and V Ă V K . Onthe other hand, if Aug K is not identically zero, then V K is 2-dimensional. But since ˜ V K is 2-dimensional and parametrized by p λ, µ q near the smooth point p , q , the image under π of theirreducible component of ˜ V K through p , q is also 2-dimensional, so it is contained in V K . Thisimplies that V Ă V K .The inclusion V Ă V K then implies that along V we have d Aug K “ h ¨p dF q for some holomorphicfunction h defined near . Therefore, if B λ Aug K ‰ h ‰ B Q Aug K B λ Aug K “ h ¨ pB Q F q h ¨ pB λ F q “ B Q F B λ F .
Theorem 1.2 then follows from Proposition 5.13. (cid:3)
Remark . According to formula (5.10), one can obtain ∆ K from the function F for every knot K . The proof of Theorem 1.2 above shows that F cuts out a smooth surface V Ă V K near . If V K is 2-dimensional (as is conjectured for every K ), then V is locally a branch of V K . Thus, evenwhen formula (1.2) does not hold, we can say that it makes sense along a suitable branch of theaugmentation variety. Remark . It is a general fact that pB µ F q| λ “ Q “ ”
0. This is because B µ F p , µ, q “ B µ ` (cid:15) ,µ pBp y p , µ qqq ˘ “ , since (cid:15) ,µ ˝ B “
0. The inclusion V Ă V K then implies that pB µ Aug K q| λ “ Q “ ” K .5.6. Independence of choices.
Writing the DGA A K with coefficients in R “ C r λ ˘ , µ ˘ , Q ˘ s required making some choices, as discussed in Section 4.1. We now study the (in)dependence ofequation (5.10), and hence of (1.2), on these choices. ‚ Capping half-disks for Reeb chords:
Let ˜ B be the new R -linear differential on A K obtained by changing the capping half-disk of a Reeb chord c . Recall that two such cappingdisks differ by an element of π p ST ˚ R , Λ K q – H p ST ˚ R , Λ K ; Z q . If c is of degree differentthan 1, then F is not altered by this change in capping disks. If c is of degree 1 and is aterm in the cycle y in Definition 4.27, then y can be suitably altered so that F (and hencealso formula (5.10)) does not change. ‚ Meridian in Λ K : A change of meridian curve for K (which we also called a change offraming (4.1)) would yield a change of variables of the form p λ, µ, Q q ÞÑ p λ, λ k µ, Q q , for UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 39 some k P Z . p λ, µ, Q q ÞÑ p λ, µλ k , Q q , Such a change would not affect the numerator on theintegrand in (5.10). The denominator would change to (cid:15) M K pB λ F ` kλ k ´ µ B µ F q . But (cid:15) M K pB µ F q “
0, as we saw in Remark 5.17, so the integrand in (5.10) remains unchanged.Note also that changing the orientation of K correponds to p λ, µ, Q q ÞÑ p λ ´ , µ ´ , Q q in F (the direction of p changes, to preserve x ¨ p “ K p µ q to ∆ K p µ ´ q .It is well-known that the Alexander polynomial is invariant under this operation (see forinstance [Sei35, Mil62]), which is compatible with the fact that ∆ K p µ q does not depend onthe orientation of K . ‚ Capping disks for longitude and meridian:
Recall that this is equivalent to a choiceof splitting of the short exact sequence (4.2). A change in that splitting corresponds to achange of variables p λ, µ, Q q ÞÑ p λQ l , µQ m , Q q , for some l, m P Z . The integrand in (5.10)changes to (cid:15) M K ˆ ´ l ´ m B µ F B λ F ´ B Q F B λ F ˙ “ ´ l ´ (cid:15) M K ˆ B Q F B λ F ˙ , where we used again the fact that (cid:15) M K pB µ F q “
0. This change has the effect of multiplying∆ K p µ q by µ ´ l , and it is compatible with the fact that the Alexander polynomial is definedup to multiplication by a power of µ .6. Examples
The right-handed trefoil.
The right-handed trefoil is represented by the 2-strand braid σ ´ .In the flow tree model, its Legendrian DGA is generated in degree 0 by orbits a , , a , , in degree1 by b , , b , , c , , c , , c , , c , and in degree 2 by e , , e , , e , , e , . The differential in degree 1 is $’’’’’’’’&’’’’’’’’% B c , “ λµ ´ ´ λµ ´ ´ p Q ´ µ q a , ´ Qa , a , B c , “ Q ´ µ ` µa , ` Qa , a , B c , “ Q ´ µ ` λµ ´ a , ` Qa , a , B c , “ µ ´ ´ Qa , ` µa , a , B b , “ λ ´ µ a , ´ a , B b , “ ´ a , ` λµ ´ a , For details, see [EN18, Section 7.2]. Note that we applied the transformation λ ÞÑ λµ ´ to theformulas in that reference, since in our conventions the x curve must be null-homologous in R z K .On chords of degree 0, we get (cid:15) M K p a , q “ µ ´ p µ ´ q , (cid:15) M K p a , q “ µ p µ ´ q . One can then compute the (cid:15) M K -linearized differential, and see that its kernel in degree 1 isspanned by y “ µ p ´ µ q c , ` µ p µ ´ q c , ` p ´ µ q b , y “ µ p µ ´ q c , ` µ p ´ µ ` µ q c , ` p ´ µ ` µ q b , y “ c , ´ c , ´ λµ ´ b , y “ λµ ´ b , ` b , Writing F “ pB y q a ÞÑ (cid:15) MK,µ p a q , we can compute pB λ F q| p λ,Q q“p , q “ µ p µ ´ qp µ ´ µ ` qpB Q F q| p λ,Q q“p , q “ µ p ´ µ ` µ ´ µ q and the right side of formula (5.10) is µ p ´ µ ` µ q , which is the Alexander polynomial of thetrefoil knot (up to the usual ambiguity of a power of µ ). Using y instead of y would also yieldthe Alexander polynomial of the trefoil. If we used y or y , then the integrand on the right side of(5.10) would be , so these cycles would not be suitable for computing the Alexander polynomial.From the DGA differential of chords of degree 1, one can also obtain the augmentation polynomialof the right-handed trefoil, which isAug K p λ, µ, Q q “ λ p µ ´ q ` λ p µ ´ µ Q ` µ Q ´ µ Q ´ µQ ` Q q ` p µ Q ´ µ Q q . See again [EN18, Section 7.2] for details. Formula (1.2) can then also be verified for the right-handedtrefoil.6.2.
Other examples.
One can find a
Mathematica notebook with augmentation polynomials ofmany knots, up to 10 crossings, on Lenhard Ng’s webpage. Equation (1.2) gives the Alexanderpolynomial for all of these examples, except the 8 knot and a connected sum of a left-handedtrefoil with a right-handed trefoil. In both cases, B x Aug K | p λ,Q q“p , q “ B Q Aug K | p λ,Q q“p , q “ V K . In the case of the connectedsum of a left-handed and a right-handed trefoil, one can understand geometrically the origin ofdifferent branches containing the line tp , µ, qu .For this, recall that the augmentation variety restricted to Q “ :(6.1) V K K | Q “ “ tp λ λ , µ q|p λ i , µ, q P V K i for i “ , u The 2-variable augmentation polynomial for the left handed trefoil is p λ ´ qp µ ´ qp ` λµ q . We have three branches: t λ “ u and t µ “ u , corresponding to the fillings M K and L K , re-spectively, as well as λ “ ´ µ ´ . The 2-variable augmentation variety for the right-handed trefoilis p λ ´ qp µ ´ qp λ ` µ q . The branches are now t λ “ u , t µ “ u and λ “ ´ µ . Equation (6.1) shows that the thirdbranches of the 2-variable augmentation varieties of the left- and right-handed trefoils combine togive a new branch covering the line λ “ Q ‰
1, we get a new branch of V K containing the line p , µ, q , as wanted.7. Disk potentials, SFT-stretching, and a deformation of the Alexanderpolynomial
In this section we discuss connections of the study undertaken in this paper to more physi-cally inspired treatments of knot theory, in particular open topological strings and Gromov–Wittentheory. We will see that our results suggest a natural Q -deformation, Q “ e t , of the Alexan-der polynomial, and that this is related to disk potentials and Floer torsion. We will also ob-serve that the higher genus counterparts give a geometric framework for invariants introduced in[GPV17, GPPV17, GM19], as outlined in [EGG ` We are assuming that V K | Q “ is equal to the 2-variable augmentation variety (the Zariski closure in the p λ, µ q -plane of the points p λ , µ , q which can be lifted to an augmentation of A K ), which is true in these examples butnot known for a general knot. UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 41 F i F j Figure 16.
Strip with two positive ends and boundary on two cotangent fibers F and F Basic disk potentials and SFT-stretching.
Given (5.10), we could define a Q -deformation, Q “ e t , of the Alexander polynomial by taking∆ K p e p , e t q “ p ´ e p q exp ˆż ´ ˆ B t F B x F ˙ ˇˇˇ t F “ u dp ˙ “ (7.1) “ p ´ e p q exp ˆż ´ ˆ B t Aug K B x Aug K ˙ ˇˇˇ t F “ u dp ˙ , where we can write the second equality when B x Aug K does not vanish. We will see how thisdeformation is related to disk potentials and Floer torsion.Consider first the case L “ L δK , for δ ą
0. Here L X R “ ∅ and we consider SFT-stretchingalong an (cid:15) -sphere bundle S (cid:15) T ˚ R for (cid:15) small compared to δ , as in [ES19]. Under such stretchingall holomorphic curves with boundary on L Y R leave R , since there are no closed Reeb orbits in ST ˚ R . Thus, after stretching the curves represent relative homology classes in H p T ˚ R z R , L q ,which is generated by x and t (recall that the meridian p is a boundary in L δ K ). In particular, count-ing generalized holomorphic disks with boundary on L in a sufficiently stretched almost complexstructure gives a Gromov–Witten disk potential W K p x, t q “ ÿ n ą C n p e t q e nx , where C n p e t q is the count of curves with boundary in the class of n times the generator of H p L q .We then have the following local parameterization of a branch of the augmentation variety:(7.2) p “ B W K B x , see [AENV14] and [Ekh18]. (One can also think of the SFT-stretched curve counts as taking placein the resolved conifold, after conifold transition.)We next show that we can define a disk potential in a similiar way also for L “ M δK , although M δ K may intersect R . To this end, we first slightly deform M δK so that its intersection with someneighborhood of the 0-section coincides with a finite number of cotangent fibers. We then SFT-stretch in a smaller (cid:15) -neighborhood. In this case the situation is less straightforward. However, thegeometry in the negative end after stretching is easy to understand. Using the flat metric we haveReeb chords γ ij between any two distinct fibers F i , F j and no other Reeb chords. Furthermore,the Lagrangian L in the negative end consists of fibers F i . Lemma 7.1.
The dimension of any moduli space of holomorphic curves with boundary on F “ Ť F i and m positive punctures equals m .Proof. The Conley–Zehnder index of the Reeb chords are 0. The dimension formula then followsfrom [CEL10]. (cid:3)
Using this formula we find that M δK defines a basic disk potential that is invariant under defor-mations. Proposition 7.2.
For sufficiently stretched almost complex structure, all holomorphic disks withboundary on M δK lie outside the (cid:15) -neighborhood. Define the corresponding basic disk potential U K p p, t q “ ÿ n ą B n p e t q e np , counting generalized holomorphic disks. This U K is invariant under deformations and (7.3) x “ B U K B p gives a local branch of the augmentation variety.Proof. We need to show that under stretching there are no broken disks with lower level in thenegative end. The minimal dimension of such a lower level is 2, corresponding to a strip with twopositive ends, see Figure 16. Since it came from a rigid disk, the strip is capped at an upper level bytwo punctured disks (the cappings might in principle consist of more complicated configurations,which would also contain components of negative index), and the sum of their expected dimensionsis ´
2, hence they do not exist generically. It follows that no disk can lie in the neighborhoodand that the count is invariant in 1-parameter families (degenerations that do not involve thenegative end are treated as usual, see [AENV14]). The last statement then follows exactly as in[AENV14]. (cid:3)
Remark . The basic potential function U K was constructed in [AENV14] for fibered knots.Proposition 7.2 extends the construction of U K to an arbitrary knot.Recall now the integral in (5.10). As we observe in Remark 5.16, the function F locally cuts outa branch of V K , along which we can write ż ´ B t F B x F dp “ ż B t x p t, p q dp. If this branch is cut out by (7.3), then(7.4) ż B t x p t, p q dp “ ż B t B p U K dp “ B t U K , hence in this case the Q -deformation of the Alexander polynomial (7.1) can be written as∆ K p e p , e t q “ p ´ e p q exp ` B t U K ˘ . We will see below that there is strong evidence in this direction for fibered knots, but not everyknot. For general K , one must also take into account disks with negative punctures at the Reebchords γ ij mentioned above, and the disk potential for M K is not unique. Remark . As we just saw, if the two branches of V K coincide, we have∆ K p e p q “ p ´ e p q lim t Ñ exp ` B t U K ˇˇ t “ ˘ . One might wonder how much information about U K p p, t q is known by ∆ K p µ q . Replacing M K and U K with L K and W K suggests that much information is lost in the limit. Using equations (4.16)and (4.17) and Proposition 4.20, the result of exchanging x and p in Theorem 1.2 is p ´ e x q exp ˆż ´ ˆ B t F B p F ˙ ˇˇˇ p e x , , q dx ˙ “ p ´ e x q exp ˆż e x ´ e x dx ˙ “ K and this expression remembers nothing about the disk potential W K , which isindeed very different for different knots. UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 43
Invariance of Floer torsion.
A first sign that input from extra negative punctures at theReeb chords γ ij may be necessary comes from the Floer cohomological torsion of CF ˚ p R , M δ K q ,defined as the product(7.6) τ K p e p , e t q “ ζ an p e p , e t q ¨ τ str p e p , e t q . Here, ζ an p e p , e t q is a count of all disconnected generalized holomorphic annuli from R to M δ K in T ˚ R . Since there are no closed Reeb orbits in ST ˚ R , generalized annuli are (trees of) disks withan intersection with the 4-chain V from (5.3), in the SFT-stretched limit, hence(7.7) ζ an p e p , e t q “ exp pB t U K p p, t qq . On the other hand, τ str p e p , e t q “ det p D q , where D is the Floer differential in the Floer cohomologycomplex CF ˚ p R , M δ K q with Novikov coefficients. This differential counts two-level curves in thestretched limit, with the lower level being a trivial strip over a straight line, as discussed above.We show in Proposition 7.2 that τ K is invariant under deformations. As discussed in Section 7.4the study of τ K leads to a more general notion of disk potential for M K , counting also disks withnegative punctures at the Reeb chords γ ij . We denote such generalized disk potentials by U ,(cid:15)K . Remark . To compute the powers of e t in counts of disks and annuli, we need to define anintersection number with the 4-chain V . To force such intersections into the interior of the curves,we must specify a way to push the boundaries of the disks in R off of R in T ˚ R , compare [EN18,Section 2]. Also, the 4-chain intersects M K so there will be contributions to e t from linking withthis locus, and defining them needs additional choices of capping paths. We will not go into thedetails of making all these choices here. Remark . In view of [EN18, Ekh18], it is natural to consider the all genus counterpart U (cid:15)K p p, t, g s q of the disk potential: U (cid:15)K p p, t, g s q “ g ´ s U ,(cid:15)K p p, t q ` U ,(cid:15)K p p, t q ` ¨ ¨ ¨ ` g ´ rs U r ` ,(cid:15)K p p, t q ` . . . , where U r ` ,(cid:15)K counts generalized holomorphic curves of Euler characteristic χ “ ´ r , possibly withnegative punctures. The Lagrangians M K and L K share ideal contact boundary Λ K . The Legen-drian SFT of Λ K determines an operator p A K p e ˆ p , e ˆ x , e t , e g s q , where the operator e ˆ p acts as multiplica-tion by e p and e ˆ x acts as e ´ g s B p . In particular, e ˆ p e ˆ x “ e g s e ˆ x e ˆ p . This is a quantization of the augmen-tation polynomial Aug K and, through holomorphic curve counting for L K , p A K p e ˆ p , e ˆ x , e t , e g s q “ p A K Ψ K “
0, where Ψ K “ exp p U (cid:15)K q . This observation together with Theorem 1.2, which inthis context gives the semi-classical limit of the un-normalized expectation value exp p N g s B t q Ψ K p p q ,lead to a large N version, or Q -deformation, of the invariant p Z , [GPV17, GPPV17, GM19], forknot complements, see [EGG ` U K gives in general a differentbranch of the augmentation variety than that associated to the Alexander polynomial. We willwrite m for a meridian curve representing the generator p of H p M K q and λ std “ ř i p i dq i for thestandard Liouville form in T ˚ R . Proposition 7.7.
The torsion τ K is invariant under deformations (changing J , Lagrangian iso-topies of M δ K , etc.) for which ş m λ std remains positive.Proof. We want to show that τ K p p, t q is invariant under deformations. Consider a generic 1-parameter family of geometric data with a perturbation scheme in which output punctures inseveral level holomorphic disks buildings are time-ordered. Using the positivity hypothesis, we canshow the invariance of τ K by studying codimension one phenomena associated to the presence ofintersection points. Bifurcations of Floer strips are associated to four types of curves (see Figure17): M δ K R η j ξ i ξ i ξ k η j ´ ξ i η k η j ´ Ñ or1 Figure 17.
Splitting of disks of index ´ ´ R M δ K Ñ M δ K R ξ i or M δ K R η j ´ Figure 18.
Pinching of holomorphic annuli into index ´ ‚ A disk of index p´ q in M p ξ i , ξ j q , ξ i ‰ ξ j . ‚ A disk of index p´ q in M p η i , η j q , η i ‰ η j . ‚ A disk of index p´ q in M p ξ i , ξ i q . ‚ A disk of index p´ q in M p η i , η i q .We note that the last two p´ q curves also arise as the limits of pinched holomorphic annuli, seeFigure 18.To see that τ K remains invariant under the first two p´ q -instances, we note that it is well-knownhow they affect the differential D p e p , e t q : by row or column operations. This leaves det p D q fixedand hence τ K does not change.Consider next a p´ q -disk δ in M p ξ i , ξ i q . Here the differential changes by adding this disk to ξ . This means that the determinant is multiplied by p ` δ q . On the other hand, the contributionto τ K from the annulus that pinched giving δ is multiplication by p ` δ q as well (consider alldisconnected curve configurations without and with this annulus). Thus the factor 1 ` δ just movesbetween the annulus count and the determinant, leaving τ K unchanged. (cid:3) Remark . Note that the deformation invariance of U K in Proposition 7.2 and of τ K in Proposition7.7, combined with (7.7), imply that in (7.6) both factors are invariant for sufficiently SFT-stretchedalmost complex structures. Another way to see this is to observe that a p´ q -disk where they wouldchange would have to limit to a Reeb chord connecting a fiber to itself, and there are no such Reebchords in ST ˚ R .7.3. Deformation of the Alexander polynomial for fibered knots.
Let K be a fibered knot.Then there is a non-zero 1-form η M on M K and we can shift M K off of R in T ˚ R . This meansthat the Floer torsion of M K can be expressed as τ K “ ζ an p e p , e t q , since there are no strips. Proposition 7.9. If K is a fibered knot and U K denotes the basic disk potential of M K then (7.8) ∆ K p e p q “ p ´ e p q lim t Ñ exp ` B t U K p p, t q ˘ . UGMENTATIONS, ANNULI, AND ALEXANDER POLYNOMIALS 45
Proof.
Combining (7.7) with Proposition 7.7, Lemma 3.8, and (2.3), we havelim t Ñ exp pB t U K p p, t qq “ lim t Ñ ζ an p e p , e t q “ ζ loop p e p q “ p ´ e p q ´ ∆ K p e p q , as wanted. (cid:3) This result, combined with (7.4) above, implies that the branch of the augmentation variety cutout by U K in (7.3) is tangent along the line p x, t q “ p , q to the branch cut out by the function F inRemark 5.16. If these branches coincided, then we would conclude the identity of Q -deformationsof the Alexander polynomial p ´ e p q τ K p e p , e t q “ ∆ p e p , e t q , where the left side is defined in (7.6) and the right side is defined in (7.1). It seems likely that thisholds for every knot K .7.4. Deformation of the Alexander polynomial for non-fibered knots.
In this section weconsider the geometry of deformations of the Alexander polynomial for non-fibered knots. This ismore complicated than the case of fibered knots and involves counts of curves with negative bound-ary punctures, which is not yet properly understood. First ideas in this direction are presented in[EGG `
20, Section 4].If K is fibered then the leading coefficient of ∆ K p e p q equals ˘
1. Consider a non-fibered knot K with this leading coefficient not equal to 1. Our first observation is that Proposition 7.9, where U K p p, t q is the basic disk potential, does not hold for K . The reason is that for action reasons,any non-constant holomorphic disk with boundary in M K contributes to the e kp -coefficient in U K where k ą
0. Therefore the leading coefficient in the right hand side of (7.8) equals 1, whereas theleading coefficient in the left hand side is not equal to 1.Since p ´ e p q τ K p e p , e t q from (7.6) gives a Q -deformation of the Alexander polynomial also in thenon-fibered case, it is natural to expect that this deformation can be expressed in terms of a diskpotential for general K . Applying SFT-stretching to (7.6), one expects corrections to U K to comefrom disks with negative punctures at the Reeb chords α ij of grading 0, connecting cotangent fibersover the points ξ i and η j . More precisely, given a collection (cid:15) of functions α ij p t, p q for all Reebchords α ij connecting ξ i to η j , we define the disk potential U ,(cid:15)K p t, p q as the count of generalizeddisks in a sufficiently SFT-stretched almost complex structure on T ˚ R with boundary on M K andnegative punctures at chords α ij , where we write a factor α ij p t, p q for each negative puncture wherethe disk is asymptotic to α ij . We then expect the following. Conjecture 7.10.
For any knot K there is a collection of functions (cid:15) “ t α ij p t, p qu such that thefollowing two properties hold (1) The disk potential U ,(cid:15)K p p, t q is invariant under deformations and x “ B U ,(cid:15)K {B p cuts out abranch of the augmentation variety. (2) The disk potential U ,(cid:15)K p p, t q recovers ∆ K p e p q as t Ñ . More precisely, lim t Ñ exp ´ B t U ,(cid:15)K p p, t q ¯ “ p ´ e p q ´ ∆ K p e p q . Condition p q in Conjecture 7.10 requires (cid:15) to be an augmentation of a dg-algebra generated byall the Reeb chords connecting cotangent fibers at the critical points ξ i and η j . For more detailson this, we refer to [EGG `
20, Section 4].
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Departamento de Matem´atica Aplicada - GMA, Instituto de Matem´atica e Estat´ıstica, Universi-dade Federal Fluminense, Rua Professor Marcos Waldemar de Freitas Reis, s/n, Bloco G - Campusdo Gragoat´a, S˜ao Domingos - Niter´oi - RJ - CEP: 24.210-201, Brazil
E-mail address : [email protected] Department of mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden and InstitutMittag-Leffler, Aurav 17, 182 60 Djursholm, Sweden
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