Tight fibred knots without L-space surgeries
Filip Misev Gilberto Spano

[
    Norman Meuschke, Vincent Stange, Moritz Schubotz, Michael Kramer, Bela Gipp
    Received ; accepted 
===============================================================================




We show there exist infinitely many knots of every fixed genus g≥ 2 which do not admit surgery to an L-space, despite resembling algebraic knots and L-space knots in general: they are algebraically concordant to the torus knot T(2,2g+1) of the same genus and they are fibred and strongly quasipositive.


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§ INTRODUCTION AND STATEMENT OF RESULT

Algebraic knots, which include torus knots, are L-space knots: they admit Dehn surgeries to L-spaces, certain 3-manifolds generalising lens spaces which are defined in terms of Heegaard-Floer homology <cit.>.


The first author recently described a method to construct infinite families of knots of any fixed genus g≥ 2 which all have the same Seifert form as the torus knot T(2,2g+1) of the same genus, and which are all fibred, hyperbolic and strongly quasipositive. Besides all the classical knot invariants given by the Seifert form, such as the Alexander polynomial, Alexander module, knot signature, Levine-Tristram signatures, the homological monodromy (in summary, the algebraic concordance class), further invariants such as the τ and s concordance invariants from Heegaard-Floer and Khovanov homology fail to distinguish these knots from the T(2,2g+1) torus knot (and from each other). This is described in the article <cit.>, where a specific family of pairwise distinct knots K_n, n∈, with these properties is constructed for every fixed genus g≥ 2.

Here we show that none of the K_n is an L-space knot (except K_0, which is the torus knot T(2,2g+1) by construction). This implies our main result:

 
For every integer g≥ 2, there exists an infinite family of pairwise distinct genus g knots K_n, n∈, with the following properties.


  * K_n is algebraically concordant to the torus knot T(2,2g+1)

  * K_n is fibred, hyperbolic and strongly quasipositive

  * K_n does not admit any nontrivial Dehn surgery to a Heegaard-Floer L-space



We show in fact that the knots K_n constructed in <cit.> do not have the same knot Floer homology as T(2,2g+1).

Below we briefly introduce the notions of L-spaces, L-space knots, quasipositivity and fibredness and relate our result to recent work by Boileau, Boyer and Gordon on the subject of L-space knots. Section <ref> contains a description of the fibred knots K_n via their monodromy (taken from <cit.>) and collects the main ingredients from Heegaard-Floer theory and Lagrangian Floer homology used in the proof of our result, which is given in Section <ref>.



 §.§ L-spaces, quasipositivity and fibredness

 are named after lens spaces, three-dimensional manifolds formed by glueing two solid tori along their boundaries. By definition, a closed 3-manifold M is an L-space if it is a rational homology sphere, that is, H_*(M,)≅ H_*(S^3,), and its Heegaard Floer homology has the smallest possible rank: (M)=|H_1(M,)|. Every lens space (except S^1 S^2, which fails to be a rational homology sphere) is in fact an L-space. More generally, 3-manifolds with finite fundamental group (the manifolds with elliptic geometry, certain Seifert fibred manifolds) are known to be L-spaces <cit.>.



A knot K⊂ S^3 is an L-space knot if some non-trivial integral Dehn surgery on K yields an L-space. Basic examples include the torus knots and the Berge knots, since they admit lens space surgeries; see <cit.>. In addition, all algebraic knots (the connected links of plane curve singularities, which include all positive torus knots) are L-space knots. This follows from a theorem of Hedden stating that certain cables of L-space knots are again L-space knots <cit.>, combined with the classical description of algebraic knots as iterated cables of torus knots, involving the Puiseux inequalities (see for example <cit.>).

By work of Ghiggini <cit.> and Ni <cit.>, <cit.>, all L-space knots are known to be fibred: they arise as the bindings of open book decompositions of S^3. In addition, Hedden proved that the open book associated to an L-space knot (or to its mirror) supports the tight contact structure of S^3, under Giroux' correspondence (see <cit.> and Ozsváth-Szabó <cit.>). In summary, L-space knots are tight fibred knots.

Besides Dehn surgery, another important construction of 3-manifolds, starting from a knot K⊂ S^3, is given by branched covering of S^3 branched along K. The n-fold cyclic branched covering, branched along K, is denoted Σ_n(K). For example, if K is an alternating knot, Σ_2(K) is always an L-space <cit.>.

Recently, Boileau, Boyer and Gordon <cit.> considered L-space knots K⊂ S^3 with the additional property that Σ_n(K) is also an L-space for some n∈, n≥ 2. They deduced strong restrictions on such knots. In order to state their results, we briefly recall the notion of strongly quasipositive knots, which were introduced and first studied by Rudolph in the 80s <cit.>. Let the symbols σ_1,σ_2,…,σ_n-1 denote the standard positive generators of the braid group on n strands, σ_i corresponding to the braid in which the i-th strand crosses over the (i+1)-st strand in the direction of the braid's orientation (and no other crossings). A braid β is called strongly quasipositive if it can be written as a product of conjugates of the σ_i:

    β=∏_j=1^d w_jσ_n_jw_j^-1,

where d∈, n_1,…,n_d∈{1,…,n-1} and the conjugating words w_j are of the special form

    w_j=σ_j-k⋯σ_j-1

for some k (depending on j). Accordingly, such braids are called strongly quasipositive. Hedden showed that a fibred knot is tight if and only if it is strongly quasipositive <cit.>. Fibredness is important here: there do exist non-fibred strongly quasipositive knots.


Let K be a strongly quasipositive knot with monic Alexander polynomial. Then


  * Σ_n(K) is not an L-space for n≥ 6.

  * If Σ_n(K) is an L-space for 2≤ n≤ 5, then K has maximal signature and its Alexander polynomial is a product of cyclotomic polynomials.



 
Let K be an L-space knot such that Σ_n(K) is an L-space for some n. Then


  * n≥ 4 implies that K is the trefoil knot.

  * n=3 implies that K is either the trefoil knot or its Alexander polynomial is equal to t^4-t^3+t^2-t+1, the Alexander polynomial of the cinquefoil knot. If it is neither the trefoil nor the cinquefoil, it is a hyperbolic knot.



These results suggest that the two properties, admitting an L-space surgery and admitting an L-space branched cover are orthogonal in the sense that only few knots seem to satisfy both properties simultaneously. When applied to genus g knots K_n satisfying the properties of Theorem <ref> (and described in the next section), the above Corollary <ref> implies that Σ_m(K_n) is not an L-space for m,g≥ 3 and n≠ 0:


Fix g≥ 3 and let K_n denote the genus g knot described below. If m≥ 3 and n≠ 0, the m-fold cyclic branched cover Σ_m(K_n) of S^3, branched along K_n is not an L-space.


For g=2, the knots K_n have the same Alexander polynomial as the cinquefoil knot T(2,5). The following (open) question was brought to our attention by Ken Baker, Michel Boileau, Marco Golla and Arunima Ray, independently. 


Is the double branched cover Σ_2(K_n) an L-space for any of the knots K_n, n≠ 0 (mentioned in Theorem <ref> and described below)? Is Σ_3(K_n) an L-space for any of the genus g=2 knots K_n?




 §.§ Acknowledgements

The authors would like to thank Michel Boileau and Paolo Ghiggini for many useful discussions. FM thanks the Max Planck Institute for Mathematics, Bonn, for its support and hospitality, where part of this work was completed. GS thanks the Laboratoire de Mathématiques Nicolas Oresme, Caen, for its support and hospitality. 




§ MONODROMIES, L-SPACES AND EXACT TRIANGLES
 



 §.§ The monodromy of the fibred knots K_n
 
Let us recall the construction of the knots K_n from the article <cit.>. Throughout, we fix an integer g≥ 2, the genus of the knots to be constructed. The fibred knots K_n are given in terms of their monodromies: K_n has a genus g fibre surface S_n. Since the topological type of S_n⊂ S^3 does not depend on n, we can identify S_n with a fixed abstract (non-embedded) surface S and consider the monodromy of K_n as a mapping class ϕ_n:S→ S. Given a simple closed curve γ⊂ S, we denote t_γ:S→ S the right Dehn twist on γ. Using this notation, the monodromy ϕ_n, n∈, is defined as the following composition of Dehn twists:

    ϕ_n := (t_β_g,n∘ t_β_g-1∘…∘ t_β_1)∘ (t_α_g∘…∘ t_α_1),

where

    β_g,n := t_c^n(β_g)

and α_i, β_j and c are the simple closed curves shown in Figure <ref>.



The curve c, shown in blue, is the boundary of a neighbourhood of α_g∪β_g-1 in S. In particular, it is nullhomologous, intersects β_g and α_g-1 in exactly two points each, and does not intersect any of the remaining curves. It follows that β_g,n∩α_g-1 consists of 4n points (see Figure <ref>). Since β_g and α_g intersect in exactly one point and α_g, c are disjoint, β_g,n∩α_g is a singleton. Moreover, all pairs of curves involved in the construction realise their minimal geometric intersection number in their homotopy classes. This is clear whenever two curves are disjoint or intersect transversely in a single point. 
For the remaining cases, use the bigon criterion <cit.>. In particular, this applies to the curves β_g,n and α_g-1, whose minimal geometric intersection number ι(α_g-1,β_g,n)=#(α_g-1∩β_g,n)=4n is used in the proof of our Theorem <ref>. These two curves are represented in Figure <ref> (the shaded band represents 2n parallel strands with alternating orientations which are part of β_g,n).


 


 §.§ Floer homology of L-space knots

The proof of our result, Theorem <ref>, relies on the following theorem by Ozsváth-Szabó, which implies that the knot Floer homology groups of an L-space knot are at most one-dimensional in each Alexander degree.

 
Let K⊂ S^3 be a knot which admits a positive integral L-space surgery. Then, there exists a sequence of integers n_0=0<n_1<…<n_k such that

    _d(S^3,K;j)≅{[                                  if (j,d)=(± n_i,δ_i), for some i;                                0                       otherwise, ].

where the supporting dimensions δ_i only depend on the n_i, according to the recursive formula 

    δ_i = {[                      0               if i = k; δ_i+1 - 2(n_i+1-n_i)+1          if k-i is odd;              δ_i+1 - 1      if k-i>0 is even. ].



We make use of two exact triangles in knot Floer homology and in Lagrangian Floer homology to bound the rank of the knot Floer homology groups of our knots K_n from below. Since the K_n are all fibred of genus g, their knot Floer homology groups in Alexander degree ± g have rank one. To prove that K_n is not L-space for n>0 we will show that

    (S^3,K_n;-g+1) > 1,

which contradicts the condition of last theorem. Note also that it does not suffice to consider the coefficients of the Alexander polynomial of the genus g knot K_n, since it equals the Alexander polynomial of the torus knot T(2,2g+1), whose coefficients are all ± 1.


It turns out that the knot Floer groups (S^3,K_n;-g+1) in the adjacent Alexander degree -g+1 are isomorphic to the symplectic homology of the monodromy ϕ_n measuring the fixed points of ϕ_n. This isomorphism, conjectured in <cit.> and proven in an upcoming article by the second author, explains our interest in this specific Alexander degree. However, our proof of Theorem <ref> does not depend on this isomorphism: we directly work with the knot Floer groups instead of passing through symplectic homology.




 §.§ Two exact triangles in knot Floer homology and in Lagrangian Floer homology

Let Y be a closed oriented 3–manifold and let K ⊂ Y be a genus g fibred knot with associated monodromy φ S → S, so that, if 𝒩 is a tubular neighbourhood of K, Y ∖𝒩 is homeomorphic to the mapping torus (S × [0,1])/((x,1) ∼ (φ(x),0)). An essential simple closed curve γ⊂ S can then be identified with a knot L_γ⊂ S ×{0}⊂ Y. 
 
Fix γ and assume that L_γ is non-trivial. Let K' ⊂ Y'  Y_-1(L_γ) and, respectively, K_0⊂ Y_0  Y_0(L_γ) be the knots determined by K in the manifolds obtained via -1 and, respectively, 0-surgery on L_γ. K' is also a genus g fibred knot, with associated fiber S and monodromy φ∘ t_γ. On the other hand K_0 is not fibred and has genus g-1. If F ⊂ Y_0 is a genus g-1 Seifert surface of K_0, then it is easy to see that (S,γ,φ(γ)) is a sutured Heegaard diagram for the sutured manifold Y_0(F) (see the definitions 2.4 and 2.10 of <cit.>). Theorem 1.5 of <cit.> and the definition of sutured Floer homology imply that

    (Y_0,K_0;-g+1) ≅(γ,φ(γ)),

 where the latter is the Lagrangian Floer homology of (γ,φ(γ)). This is a homology whose generators of the chain complex are intersection points of γ and φ(γ) (which are assumed to be transverse; see for example <cit.> for the details of the definition). We recall also that if two curves a and b are homotopic then
 
    (a,b)=2

 and if they are not homotopic then 
 
    (a,b) = ι(a,b),

where ι(a,b) denotes the geometric intersection number of the curves a and b.

 
There is an exact triangle

    (Y,K;-g+1) [rr]       (Y',K';-g+1) [dl] 
       (γ,φ(γ)). [ul]




This is a direct consequence of (<ref>) and Ozsváth and Szabó's exact sequence for knot Floer homology.


The second lemma we are going to use is a special case of a theorem due to Seidel. It generalises the well-known identity ι(t_a(b),b)=ι(a,b)^2 for essential simple closed curves a,b on a surface <cit.>.

 
Let S be a compact oriented surface with boundary and t_c:S→ S the Dehn twist on a simple closed curve c⊂ S. For any pair of simple closed curves a,b⊂ S, there is an exact triangle of Lagrangian Floer cohomology groups 

    (t_c(a),b) [rr]       (a,b) [dl] 
       (c,a) ⊗(b,c) [ul]





§ PROOF OF THE MAIN THEOREM
 

Let ψ t_β_g-1∘…∘ t_β_1∘ t_α_g∘…∘ t_α_1, so that ϕ_n=t_β_g,n∘ψ. Let (Y,K) be given by the open book (S,ψ). That is, K⊂ Y is a fibred knot with monodromy ψ:S→ S. Note that both the 3-manifold Y and the knot K⊂ Y are independent of n, since S only depends on g and ψ is also independent of n. Now consider the open book (S^3,K_n) associated to the fibred knot K_n, which is the knot of interest. It is obtained from (Y,K) by (-1)-Dehn surgery along the curve β_g,n⊂ S⊂ Y. We can therefore apply Lemma <ref> to this situation, where (Y',K')=(S^3,K_n).

    (Y,K;-g+1) [rr]^x       (S^3,K_n;-g+1) [dl]^y 
       (β_g,n,ψ(β_g,n)) [ul]^z

Exactness and the rank-nullity formula for x,y and z imply

    (S^3,K_n;-g+1)    =    y +  y 
       =    x +  z 
       =   (Y,K;-g+1) -  x 
          + (β_g,n,ψ(β_g,n)) -  z 
       ≥   (β_g,n,ψ(β_g,n)) - (Y,K;-g+1)

To calculate the rank of (Y,K;-g+1), we apply Lemma <ref> again, now taking (Y',K') to be the open book obtained from (Y,K) by (-1)-Dehn surgery along β_g,0=β_g⊂ S⊂ Y. We obtain Y'=S^3 and K'=K_0. The exact triangle from Lemma <ref> now reads as follows.

    (Y,K;-g+1) [rr]       (S^3,K_0;-g+1) [dl] 
       (β_g,ψ(β_g)) [ul]

Since K_0=T(2,2g+1), we know that (S^3,K_0;-g+1)=1. Further, the curve ψ(β_g)=t_β_g-1(t_α_g(β_g)) intersects β_g in exactly one point, whence (ψ(β_g),β_g)=ι(β_g,ψ(β_g))=1. Because the above triangle is exact, we deduce 

    (Y,K;-g+1) ≤ 2.


The missing piece of information is (β_g,n,ψ(β_g,n)). To compute it, observe first that

    ψ(β_g,n)=t_β_g-1∘ t_α_g∘ t_α_g-1(β_g,n),
 because β_g,n=t_c^n(β_g) does not intersect any of the curves α_i,β_i for i≤ g-2. We could, in principle, directly compute the intersection number by studying the curves β_g,n and ψ(β_g,n) on the surface S. But Lemma <ref> conveniently helps to simplify the calculation. First let us apply it to the curves (a,b,c)=(β_g,n,α_g,α_g-1), which gives 

    (t_α_g-1(β_g,n),α_g)=(β_g,n,α_g) = 1

because α_g-1∩α_g = ∅ (so that the lower term of the exact triangle has rank 0) and β_g,n and α_g intersect in a single point.


Now apply Lemma <ref> three times to estimate (β_g,n,ψ(β_g,n)) = (ψ(β_g,n),β_g,n):



  1. First apply the lemma to (a,b,c)=(t_α_gt_α_g-1(β_g,n),β_g,n,β_g-1). Since β_g-1∩β_g,n=∅, the lower corner of the exact triangle vanishes and we get:

    (ψ(β_g,n),β_g,n)=(t_α_g∘ t_α_g-1(β_g,n),β_g,n).



  2. Next choose (a,b,c)=(t_α_g-1(β_g,n),β_g,n,α_g). The lower term in the exact triangle is (α_g,t_α_g-1(β_g,n)) ⊗(β_g,n,α_g) which has rank 1 by (<ref>). The exactness of the triangle implies that:

    (t_α_g∘ t_α_g-1(β_g,n),β_g,n) ≥(t_α_g-1(β_g,n),β_g,n) - 1.



  3. Finally, choose (a,b,c)=(β_g,n,β_g,n,α_g-1). The upper right term in the corresponding triangle is (β_g,n,β_g,n) which has rank 2 by equation (<ref>). It follows that:

    (t_α_g-1(β_g,n),β_g,n)    ≥   ((α_g-1,β_g,n) ⊗(β_g,n,α_g-1)) - 2
        =    ((α_g-1,β_g,n))^2 -2.


Summing up we get

    (ψ(β_g,n),β_g,n) ≥ ((α_g-1,β_g,n))^2 -3.


On the other hand the computation in Section <ref> gives (α_g-1,β_g,n) = ι(α_g-1,β_g,n) =4n, so that:

    (ψ(β_g,n),β_g,n) ≥ 16n^2 -3.


Finally, substituting the last estimation and the one in (<ref>) in the rank inequality obtained at the beginning of this section we get:

    (S^3,K_n;-g+1)    ≥   (β_g,n,ψ(β_g,n)) - (Y,K;-g+1) 
       ≥    16n^2-5

If n ≠ 0 this quantity is strictly grater than 1 and Theorem <ref> implies that K_n cannot be an L-space knot. This establishes the third property stated in the Theorem. Properties (1) and (2) are proven in <cit.>.




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