Nonfillable Legendrian knots in the 3-sphere
Tolga Etgü


LOW-MASS WHITE DWARFS WITH HYDROGEN ENVELOPES
 AS A MISSING LINK IN THE TIDAL DISRUPTION MENU
    Enrico Ramirez-Ruiz
    December 30, 2023
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Let Λ be a Legendrian knot in the standard contact 3-sphere. 
If Λ bounds an orientable exact Lagrangian surface Σ in the standard symplectic 4-ball, then the genus of S is equal to the slice genus of (the smooth knot underlying) Λ, the rotation number of Λ is zero as well as the sum of the Thurston-Bennequin number of Λ and the Euler characteristic of Σ, and moreover the linearized contact homology of Λ with respect to the augmentation induced by Σ is isomorphic to the (singular) homology of Σ. 
It was asked in <cit.> whether the converse of this statement holds. 
We give a negative answer to this question providing a family of Legendrian knots with augmentations which are not induced by exact Lagrangian fillings although the associated linearized contact homology is isomorphic to the homology of the smooth surface of minimal genus in the 4-ball bounding the knot.





§ INTRODUCTION


Let Λ be a Legendrian knot in the standard contact 3-sphere. 
If Λ bounds an  exact orientable Lagrangian surface Σ in the standard symplectic 4-ball, then the genus of Σ is equal to the slice genus of (the smooth knot underlying) Λ, the rotation number of Λ is zero as well as the sum of the Thurston-Bennequin number of Λ and the Euler characteristic of Σ by a theorem of Chantraine <cit.>. Moreover the linearized contact homology of Λ with respect to the augmentation induced by Σ is isomorphic to the (singular) homology of Σ by a theorem of Seidel <cit.>. 
Question (8.9) in <cit.> asks whether every augmentation for which Seidel's isomorphism holds is induced by a Lagrangian filling.
We give a negative answer to this question based on
the family of Legendrian knots

    {Λ_p,q,r,s : p,q,r,s ≥ 2,  p ≡ q ≡ r+1 ≡ s+1  2 }

given by the  Lagrangian  projection in Fig. (<ref>). 
Throughout the paper we will always be working under the above assumption that the parities of p and q match and they are the opposite of those of r and s.
The rotation and Thurston-Bennequin numbers of Λ_p,q,r,s are 0 and 5, respectively. 
This gives a lower bound of 3 on the slice genus. 
On the other hand, the Seifert surface we obtain from this projection has genus 3 hence both the slice genus and the Seifert genus are 3.

The Chekanov-Eliashberg dg-algebra of Λ_p,p,p+1,p+1 admits an augmentation which is not induced by any exact orientable Lagrangian filling, although the corresponding linearized contact homology is isomorphic to the  homology of a surface of genus 3 with one boundary component.

Our examples are inspired by a deformation argument in <cit.> related to the Chekanov-Eliashberg algebra where the Legendrian link of unknots linked according to the D_4 tree is shown to be significant, specifically over a base field of characteristic 2 (see the last part of the proof of Thm.(14) in <cit.> and Rem. (<ref>) below). 
As can be seen in Fig. (<ref>), Λ_p,q,r,s is constructed from the D_4 link by adding twists that turn it into a Legendrian knot. 
A similar construction was previously used in <cit.> on a different link to produce infinite families of Legendrian knots not isotopic to their Legendrian mirrors.

In the next section we describe the Chekanov-Eliashberg algebra and compute the linearized contact homology of our examples. 
In Sec. (<ref>) we gather enough information on the strictly unital A_∞-algebras obtained by dualizing the Chekanov-Eliashberg algebras of our Legendrian knots to prove that they are not quasi-isomorphic to the A_∞-algebra of cochains on a closed surface whenever the base field has characteristic 2.  This proves the nonfillability of our examples by a duality result of Ekholm and Lekili from their recent preprint <cit.> (see Thm. (<ref>) below).

Acknowledgments. We would like to thank Yankı Lekili and Joshua Sabloff for discussions on the subject and comments on a draft of this paper. It is a pleasure to thank Princeton University for the  hospitality. This research is partially supported by the Technological and Research Council of Turkey through a BIDEB-2219 fellowship.



§ LINEARIZED CONTACT HOMOLOGY  OF Λ_P,Q,R,S


The Chekanov-Eliashberg algebra of a Legendrian knot is a differential graded algebra generated by Reeb chords from the knot to itself.
We refer to <cit.> for the combinatorial description of the Chekanov-Eliashberg algebra of a Legendrian knot in the standard contact 3-sphere. 

We denote the Chekanov-Eliashberg algebra of Λ= Λ_p,q,r,s over a field  by (L, )̣. 
Since the rotation number of Λ is 0, we have a -grading on (L, )̣. 




The generators of (L, )̣ are as indicated in Fig. (<ref>):  

    a^x_0, … , a_p^x ,  a^y_0,  … , a_q^y,     a^z_0, … , a_r^z,  a^w_0, … , a_s^w 
    
    x_0 , … , x_p,   y_0 , … , y_q,      z_0 , … , z_r,   w_0 , … ,  w_s 
    
    a_0 , b_1,    … , b_6

with gradings

    |*_i|=0, |a_0| =|a^*_i|=1 ,   *  ∈{ x,y,z,w }
    
    |b_1|=-|b_4|    = p-r+1,  |b_2|=-|b_5| = q-r+1,  |b_3|=-|b_6| = r-s

Let : L → be the augmentation which maps all *_i to -1 for *  ∈{ x,y,z,w }.

There is an isomorphism 

    HC_*^ (Λ_p,p,p+1,p+1) ≅  H_1-* (Σ ; )

between the linearized contact homology of Λ_p,p,p+1,p+1 with respect to the augmentation  and the homology of the orientable surface Σ of genus 3 with one boundary component.


Counting the relevant immersed polygons with the choice of a base-point on Λ_p,q,r,s as indicated by ∙ in Fig. (<ref>) we see that nontrivial differentials are given by

    ạ_0     = 1 - w_sz_ry_qx_p 
    ạ^x_0     = 1 + x_0 + b_1b_4 
    ạ^y_0     = 1 + y_0 + b_2b_5 
    ạ^z_0     = 1 + z_0 + b_4b_1+ b_5b_2 + z_0 b_6b_3 +  b_4b_1 b_5b_2  
    ạ^w_0     = 1 + w_0 + b_3b_6 
    ạ_i^*     = 1 - *_i-1*_i 
     *  ∈{ x,y,z,w }  i ≥ 1
 
Conjugating $̣ by the automorphismid+gives another differential^̣onL^̣ a_0     = w_s + z_r + y_q+ x_p - w_sz_r- w_sy_q - w_sx_p - z_ry_q - z_sx_p - y_qx_p 
        + w_sz_ry_q+ w_sz_rx_p+w_sy_qx_p+z_ry_qx_p- w_sz_ry_qx_p 
    ^̣ a^x_0     =  x_0 + b_1b_4 ,   ^̣ a^y_0 = y_0 + b_2b_5 ,   ^̣ a^w_0  =  w_0 + b_3b_6 
    ^̣ a^z_0     =z_0 + b_4b_1+ b_5b_2 - b_6b_3 + z_0 b_6b_3 +  b_4b_1 b_5b_2  
    ^̣ a_i^*     =  *_i-1+*_i-*_i-1*_i  ,   *  ∈{ x,y,z,w }  i ≥ 1


Applying the elementary transformationa_0 ↦  a_0 -(-1)^p(∑_i=0^p (-1)^i a_i^x +  ∑_i=0^q (-1)^i a_i^y
- ∑_i=0^r (-1)^i a_i^z- ∑_i=0^s (-1)^i a_i^w)simplifies the computation of linearized contact homologyHC_*^(Λ)ofΛassociated to the augmentationand more importantly, the description of theA_∞-algebras that will be discussed in the next section. 

At this point, we have the following presentation of the differential_̣1^on the linearized complex which computesHC_*^(Λ_p,q,r,s):

    _̣1^ a_0  = _̣1^ b_j  = 0  
    _̣1^ a^x_0  =  x_0  ,  _̣1^ a^y_0    = y_0,    _̣1^ a^z_0  =z_0,     _̣1^ a^w_0  =  w_0  
    _̣1^ a_i^*  =  *_i-1+*_i     ,   *  ∈{ x,y,z,w }  i ≥ 1

It is clear thatHC_*^(Λ_p,q,r,s)is spanned bya_0 , b_1, …, b_6. 
Moreover, ifp=q=r-1=s-1, then|b_i|=0for alliand we get the graded isomorphism in the statement.


§ THE AUGMENTATION  IS NOT INDUCED BY A LAGRANGIAN FILLING


In this section, we prove thatΛ_p,q,r,shas no exact Lagrangian filling associated to the augmentationby using a result from a recent preprint of Ekholm and Lekili. 

The following is a consequence of  <cit.>.

If Λ has an exact Lagrangian filling Σ, then there is an A_∞ quasi-isomorphism between RHom_CE^*(, ) and the A_∞-algebra C^*(S;) of (singular) cochains on the closed surface S obtained by capping the boundary of Σ, where CE^*=L_-* and  is equipped with a CE-module structure by the augmentation _Σ : CE → induced by the filling Σ.


In order to describe theA_∞-algebraRHom_CE^*(, )in the above statement, we  utilize the isomorphism betweenRHom_CE^*(, )and the linear dualof the  LegendrianA_∞-coalgebra defined in the more general setting of <cit.>. 
In the current setup, the strictly unitalA_∞-algebracan be obtained from the non-unitalA_∞-algebra on the linearized cochain complex defined in <cit.> (which is also the endomorphism algebra ofin theAug_-category of <cit.>) by adding a copy ofto make it unital (cf. <cit.>). 

The description of≅RHom_CE^*(, )we provide is based on the presentation of(L, ^̣)obtained at the end of the proof of Prop. (<ref>).
We abuse the notation and denote the duals of the generators ofLby the generators themselves.
The nontrivialA_∞-products on(besides those dictated by strict unitality)  areμ_^1 (*_i)  =a_i^* + a_i+1^*  ,   *  ∈{ x,y,z,w }  i ≥ 0μ_^1 (x_p)  =a_p^x   ,  μ_^1 (y_q)  =a_q^y   ,  μ_^1 (z_r)  = a_r^z   ,  μ_^1 (w_s)  =a_s^wμ_^2 (b_1,b_4) = (-1)^p+1a_0+a^x_0  ,  μ_^2 (b_2,b_5) =(-1)^p+1 a_0+a^y_0  ,  μ_^2 (b_3,b_6) = (-1)^pa_0+a^w_0μ_^2 (b_4,b_1) = μ_^2 (b_5,b_2) = (-1)^pa_0+a^z_0  ,  μ_^2 (b_6,b_3) =(-1)^p+1a_0-a^z_0μ_^2 (x_i-1, x_i) = (-1)^p+ia_0 - a_i^x  ,    μ_^2 (y_i-1, y_i) = (-1)^p+ia_0 - a_i^yμ_^2 (z_i-1, z_i) = (-1)^p+i+1a_0 - a_i^z,    μ_^2 (w_i-1, w_i) = (-1)^p+i+1a_0 - a_i^wμ_^2 (w_s,z_r) = μ_^2 (w_s,y_q) = μ_^2 (w_s,x_p) = μ_^2 (z_r,y_q)  = μ_^2 (z_r,x_p)  =μ_^2 (y_q,x_p) = - a_0μ_^3 (z_0 , b_6,b_3) = a_0+a^z_0μ_^3 (w_s,z_r,y_q) = μ_^3 (w_s,z_r,x_p) =  μ_^3 (w_s,y_q,x_p) = μ_^3 (z_r,y_q,x_p) = a_0μ_^4 (b_4,b_1,b_5,b_2) = a_0+a^z_0  ,  μ_^4 (w_s,z_r,y_q,x_p) = - a_0and the gradings inare|a_0| =|a^*_i|=2 ,  |*_i|=1  *  ∈{ x,y,z,w }  i ≥ 0|b_1|=-|b_4| = p-r+2,  |b_2|=-|b_5| = q-r+2,  |b_3|=-|b_6| = r-s+1A homological perturbation argument, as suggested by Prop. (1.12) and Rem. (1.13) in <cit.>, provides a minimal model(, μ^∙_)quasi-isomorphic to(, μ^∙_). 
From the above description of theA_∞-products we see the decomposition=⊕, whereis generated by{ 1, a_0 , b_i : i=1,… , 6}and gives a minimal model for, whereasis the subalgebra generated by the rest of the generators ofand acyclic with respect to the differentialμ^1.  
We choose a contracting homotopyT^1: →withμ^1_T^1 + T^1 μ^1_= F^1G^1 - idas followsT^1 (a_i^x) = -x_i +x_i+1+ ⋯ +(-1)^p-i+1 x_p ,  T^1 (a_i^y) =- y_i +y_i+1+  ⋯ + (-1)^q-i+1y_q
    T^1 (a_i^z) = -z_i +z_i+1+ ⋯ +(-1)^r-i+1 z_r ,  T^1 (a_i^w) = - w_i +w_i+1+ ⋯ + (-1)^s-i+1w_sThis homotopyT^1and Eqn. (1.18) in <cit.> suffices to compute theA_∞-productsμ^∙_on. 
    To begin with, the only nontrivialμ^2_products (besides those dictated by strict unitality) areμ_^2(b_1,b_4)=μ_^2(b_2,b_5)=μ_^2(b_6,b_3)=(-1)^p+1a_0μ_^2(b_4,b_1)=μ_^2(b_5,b_2) = μ_^2(b_3,b_6)=(-1)^pa_0Moreover,μ^3_vanishes since 
    
      * μ^3_ is trivial on ⊗⊗, and 
    
      * μ^2_ vanishes on im(F^2) ⊗⊂ im(T^1) ⊗⊂⊗  and on ⊗ im(F^2) ⊂⊗ 
    whereF^2 = T^1 ∘μ^2_.
    
    Proceeding further, Eqn. (1.18) in <cit.> ford=4simplifies asμ^4_ (α_4, … , α_1) = G^1 (μ^4_ (α_4, … , α_1)+ μ^3_(F^2 (α_4, α_3) , α_2 , α_1)+μ^2_(F^2 (α_4, a_3) , F^2 (α_2 , α_1)))whereG^1 : →is the projection. 
    
    At this point, one immediately sees that all three summands on the right hand side of the above formula forμ^4_vanish on quadruples which are not of the form(b_i,b_i±3,b_j,b_j± 3).  
    In order to prove the key proposition below, it suffices to computeμ^4_(b_i,b_j,b_k,b_l), where(b_i,b_j,b_k,b_l)is a cyclic permutation of(b_1,b_2,b_5,b_4),(b_1,b_5,b_2,b_4), or(b_2,b_5,b_2,b_5).
    To this end, it is straightforward, if tedious, to check that, depending on the parity ofp, the following are the only nontrivial ones among these tenμ^4products:μ^4_(b_4,b_1,b_2,b_5)=a_0,  μ^4_(b_5,b_2,b_4,b_1)=μ^4_(b_5,b_2,b_5,b_2)=-a_0if pis even, andμ^4_(b_4,b_1,b_2,b_5)=μ^4_(b_4,b_1, b_5, b_2)=a_0,  μ^4_(b_2,b_5,b_2, b_5)=-a_0ifpis odd.
    
    Besides the computations above, another important ingredient for the proof of the proposition below is the following formality statement.
    
    Over a field  of characteristic 2, the algebra of cochains C^*(S;) on a closed orientable surface S is a formal differential graded algebra.
    
    Over the base field=, there is the classical (and much more general) formality result in <cit.>.
    Since we were not able to locate an extension of this result to nonzero characteristic cases in the literature, we provide a proof of Lem. (<ref>)  at the end of this section.
    In fact, the characteristic condition in the statement of Lem. (<ref>) can be removed by a straightforward modification of the proof.
    
    
    If the characteristic of the base field  is 2, then the A_∞-algebra  is not A_∞ quasi-isomorphic to the algebra of cochains on a closed orientable surface S.
    
    
    By Lem. (<ref>), it suffices to prove that there is no A_∞ quasi-isomorphism between  and H^*(S;).
    Suppose that  is an A_∞-algebra homomorphism from  to  H^*(S; ). 
    Since  is a minimal A_∞-algebra, μ^1_ is trivial. 
    We have also established above that μ^3_ vanishes as well. 
    As a consequence, a particular set of A_∞-functor equations, satisfied by the family of graded multilinear maps ^d : ^⊗ d→ H^*(S;)[1-d], simplifies to 
    
        ^1  ( μ^4_ (b_i, b_j, b_k,b_l) )   
           =  ^3 (b_i, b_j, b_k) ∪^1(b_l) +  ^1(b_i) ∪^3 ( b_j, b_k, b_l) +  ^2(b_i, b_j) ∪^2 (  b_k, b_l)
           
         + ^3  (μ_^2 (b_i, b_j) , b_k , b_l) + ^3(b_i, μ_^2 (b_j , b_k ), b_l) 
        + ^3(b_i, b_j , μ_^2 (b_k , b_l))
    
    
    In the rest of the proof we refer to the above equation as Eq_(i,j,k,l)
    and consider the sum of all the equations Eq_(i,j,k,l) where (i,j,k,l) is a cyclic permutation of (1,2,5,4), (1,5,2,4) or (2,5,2,5). 
    
    First of all, the computation preceding this proposition implies that the sum of the left hand side of these ten equations is equal to (-1)^p+1^1(a_0).
    In contrast, the right hand side of the sum of these ten equations is 0. 
    Once we establish this, we get ^1(a_0)=0 and hence  is not a quasi-isomorphism.
    
    To prove the vanishing of the right hand side we consider the terms on the right hand side in three separate groups and argue that each group adds up to 0 under the assumption that char()=2. First observe that, since the cup product ∪ is (graded-)commutative, each of the first two terms on the right hand side of the equation Eq_(i,j,k,l) appears in exactly one other equation, namely Eq_(l,i,j,k) or Eq_(j,k,l,i) . 
    For the same reason, the third term on the right hand side of Eq_(i,j,k,l) is cancelled by that of Eq_(k,l, i,j), unless of course (i,j)=(k,l). 
    This leaves us with the sum of the third terms of Eq_(2,5, 2,5) and Eq_(5, 2,5,2),
    ^2(b_2, b_5) ∪^2 (  b_2, b_5) + ^2(b_5, b_2) ∪^2 (  b_5, b_2)
    which vanish by the general properties of the cup product.
    Finally, remember that we always have μ^2_  (b_i , b_j)=0 for |i-j|≠ 3 and, if char()=2, 
    μ^2_  (b_4 , b_1) = μ^2_  (b_2 , b_5) = μ^2_  (b_5 , b_2)=a_0  . 
    This suffices to conclude that each of the last three terms on the right hand side of any one of the equations is either 0 or it appears in exactly two of our equations, e.g. the fifth term in Eq_(2,4,1,5) is equal to the fifth term in Eq_(2,5,2,5) .
    
    The Legendrian knot Λ_p,q,r,s admits an augmentation which is not induced by an exact orientable Lagrangian filling.
    
    When char ()≠2 our proof of Prop. (<ref>) breaks down because the right hand side of the sum of the ten equations we consider in the last step of the proof is equal to
     2 (-1)^p ( ^3  (a_0 , b_2 , b_5) + ^3(b_2, a_0, b_5) + ^3(b_2, b_5 , a_0))
    which is not necessarily 0 in general.
    (of Lem. (<ref>))
    We prove the formality of the differential graded algebra C=C^*(S, ) of (simplicial) cochains with the cup product on the closed surface S associated to the triangulation given in Fig. (<ref>) by providing a zig-zag of explicit dg-algebra quasi-isomorphisms connecting C and the cohomology algebra H=H^*(S, ) of S. 
    
    
    
    We denote the generators of C by
    e_i, α_j, β_j, θ_k, γ_k     1≤ j ≤ g ,  1≤ k ≤ 4g 
    which represent the duals of the simplicies
    e^0_i, a_j, b_j , e^1_k, e^2_k  
    as indicated in Fig. (<ref>). 
    The nontrivial differentials and products can be read from the triangulation as
    ẹ_1  = ẹ_2 = θ_1 + ⋯ + θ_4gα̣_j  = γ_4j-3+γ_4j-1 ,   β̣_j  = γ_4j-2+γ_4j ,   θ̣_k  = γ_k-1 +γ_k 
    and
    e_ie_i=e_i,   e_1θ_k = θ_k ,   e_1γ_k = γ_k,   θ_k e_2 = θ_k ,   γ_k e_2 = γ_k e_2 α_j = α_j e_2 = α_j ,   e_2 β_j = β_j e_2 = β_jθ_4j-3α_j=γ_4j-3 ,   θ_4j-2β_j=γ_4j-2θ_4jα_j=γ_4j-1 ,   θ_4j+1β_j=γ_4j
    (In the above equations and the rest of the proof, indices should always be interpreted modulo 4g.)
    
    We now define another dg-algebra, quasi-isomorphic to C and with a simplified differential so that the rest of the proof is more transparent. 
    This new dg-algebra C' is generated by 
    e,  φ_j, ψ_j, ν, _1, ζ_1, ξ_l,  ν_l       1≤ j ≤ g ,  1 ≤ l ≤ 4g-1 
    so that the map Φ : C' → C
    defined by 
    
        Φ:          e ↦ e_1 + e_2  ,     _1 ↦ e_1,    ζ_1 ↦θ_1 + ⋯θ_4g
           φ_j ↦α_j + θ_4j-2 + θ_4j-1,     ψ_j ↦β_j + θ_4j-1 +θ_4j , 
           ξ_l ↦θ_l  ,    ν_l ↦γ_l-1 + γ_l,    ν↦γ_4g
    
    is a dg-algebra quasi-isomorphism.
    More precisely, on C', the nontrivial differentials are 
    '̣_1 = ζ_1,   '̣ξ_l = ν_l e is the identity element, and the remaining products are 
    
        φ_j ψ_j = ν + ν_1 + ⋯ + ν_4j-2 ,      ψ_j φ_j = ν + ν_1 + ⋯ + ν_4j-1
        ξ_4j-3φ_j=ν + ν_1 + ⋯ + ν_4j-3 ,       ξ_4j-2ψ_j=ν + ν_1 + ⋯ + ν_4j-2
         j<g,   ξ_4jφ_j=ν + ν_1 + ⋯ + ν_4j-1,       ξ_4j+1ψ_j=ν + ν_1 + ⋯ + ν_4j ,    ξ_1ψ_g=ν
        _1_1=_1,   _1ξ_l = ξ_l ,      _1φ_j = ξ_4j-2 +ξ_4j-1
         j<g,    _1ψ_j = ξ_4j-1 +ξ_4j,      _1ψ_g= ζ_1 + ξ_1 + ⋯ξ_4g-2,   _1ν = ν
          _1ζ_1 = ζ_1,   _1ν_l = ν_l,     ζ_1φ_j= ν_4j-2 + ν_4j-1
         j<g,    ζ_1ψ_j=ν_4j-1+ν_4j ,      ζ_1ψ_g= ν_1 + ⋯ + ν_4g-2
    
    
    In the next step, we define yet another dg-algebra C by stabilizing C', i.e. C contains C' as a subalgebra and the inclusion map is a dg-algebra quasi-isomorphism. 
    Namely, we add the generators 
    _k, ζ_k       2≤ k ≤ 2g+1 
    with |_k|=0 ,  |ζ_k|=1   _k = ζ_k ,  ζ_k = 0
    to those of C'.
    and extend the algebra structure to C by adding the following nontrivial products
    
        _2jψ_j = ξ_1 + ⋯ + ξ_4j-2,      _2j+1φ_j = ξ_1 + ⋯ + ξ_4j-1
        ζ_2jψ_j = ν_1 + ⋯ + ν_4j-2,      ζ_2j+1φ_j = ν_1 + ⋯ + ν_4j-1
    
    for j=1, … , g.
    
    Finally, it is clear that the map Φ : H →C defined on the cohomology algebra H=H^*(S; ) by 
    Φ :   e ↦ e ,   φ_j ↦φ_j + ζ_2j ,   ψ_j ↦ψ_j + ζ_2j+1 ,   ν↦ν
    is a dg-algebra quasi-isomorphism
    proving the formality of C=C^*(S; ).
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