Contact (+1)-surgeries along Legendrian Two-component Links
In this paper, we prove that the contact Ozsváth-Szabó invariant of a contact 3-manifold vanishes if it can be obtained from the standard contact 3-sphere by contact -surgery along a Legendrian two-component link with the linking number of and being nonzero and satisfying . As a corollary, the contact Ozsváth-Szabó invariant of the contact 3-manifold obtained from the tight contact by contact -surgery along a homologically essential Legendrian knot vanishes for any positive integer . We also prove that the contact Ozsváth-Szabó invariant of a contact 3-manifold vanishes if it can be obtained from the standard contact by contact -surgery along a Legendrian Whitehead link. In addition, we give a sufficient condition for the contact 3-maniold obtained from the standard contact 3-sphere by contact -surgery along a Legendrian two-component link being overtwisted.
2000 Mathematics Subject Classification:
A contact structure on a smooth oriented 3-manifold is a smooth tangent 2-plane field such that any smooth 1-form locally defining as satisfies the condition . A contact structure is coorientable if and only if there is a global 1-form with . Throughout this paper, we will assume our 3-manifolds are oriented, connected and our contact structures are cooriented. A contact structure on is called overtwisted if one can find an embedded disc in such that the tangent plane field of along its boundary coincides with ; otherwise, it is called tight. Any closed oriented 3-manifold admits an overtwisted contact structure (cf. ). It is much harder to find tight contact structures on a closed oriented 3-manifold. The following question is still open: Which closed oriented 3-manifolds admit tight contact structures?
One way of obtaining new contact manifolds from the existing one is through contact surgery. Suppose is a Legendrian knot in a contact 3-manifold , i.e., is tangent to the given contact structure on . Contact surgery is a version of Dehn surgery that is adapted to the contact category. Roughly speaking, we delete a tubular neighborhood of , then reglue it, and obtain a contact structure on the surgered manifold by extending from the complement of the tubular neighborhood of to a tight contact structure on the reglued solid torus (see  for details). In , the first author and Geiges proved that every closed contact 3-manifold can be obtained by contact -surgery along a Legendrian link in , where denotes the standard contact structure on .
Heegaard Floer theory associates an abelian group to a closed, oriented Spin 3-manifold , and a homomorphism
to a Spin cobordism between two Spin 3-manifolds and . Write for the direct sum over all Spin structures on and for the sum over all Spin structures on . Throughout this paper, we work with the Heegaard Floer homology with coefficients in . In , Ozsváth and Szabó introduced an invariant for any closed contact 3-manifold . We call it the contact Ozsváth-Szabó invariant, or simply the contact invariant of . It is shown that if is overtwisted , and if is strongly symplectically fillable . If the contact manifold is obtained from by contact -surgery along a Legendrian knot , then we have
where stands for the cobordism induced by the surgery with reversed orientation. This functorial property of the contact invariant can be proved by an adaption of [23, Theorem 4.2] (cf. [15, Theorem 2.3]).
It is natural to ask whether the contact invariant of a contact 3-manifold obtained by contact -surgery along a Legendrian link is trivial or not. Here, contact -surgery along a Legendrian link means contact -surgery along each component of the Legendrian link. Most known results concern contact -surgeries along Legendrian knots. In , Lisca and Stipsicz showed that contact -surgeries along certain Legendrian knots in yield contact 3-manifolds with nonvanishing contact invariants for any positive integer . In , Golla considered contact 3-manifolds obtained from by contact -surgeries along Legendrian knots, where is any positive integer. He gave a necessary and sufficient condition for the contact invariant of such a contact 3-manifold to be nonvanishing. In , Mark and Tosun extended Golla’s result to contact -surgeries, where is rational.
In this paper, we study contact -surgeries along Legendrian two-component links in . Our main result is:
Suppose is a Legendrian two-component link in the standard contact 3-sphere whose two components have nonzero linking number. Assume satisfies , where denotes the mirror of . Then contact -surgery on along yields a contact 3-manifold with vanishing contact invariant.
The main tool for proving this theorem is a link surgery formula for the Heegaard Floer homology of integral surgeries on links developed by Manolescu and Ozsváth . Here, is a numerical invariant defined by Hom and the third author in  based on work of Rasmussen . It is shown in [10, Proposition 3.11] that a knot satisfies the condition if and only if we have a filtered chain homotopy equivalence
Below, we list some interesting families of knots that satisfy the condition
of Theorem 1.1.
The most basic examples are the slice knots.
We are particularly interested in Legendrian slice knots with Thurston-Bennequin invariant , as contact -surgeries along these knots result in contact 3-manifolds with nonvanishing contact invariants . Nontrivial knot types of smoothly slice knots with at most 10 crossings that have Legendrian representatives with Thurston-Bennequin invariant are (the mirror of ) and . Moreover, recall that , where denotes the Legendrian connected sum of the Legendrian knots and . By performing Legendrian connected sums, we obtain infinitely many Legendrian slice knots which have Thurston-Bennequin invariant equal to .
In Figures 1, 2 and 3 below, we give several Legendrian two-component links in that include a Lengendrian unknot and a Legendrian knot of type , respectively. Note that one obtains nonvanishing contact invariant after performing contact -surgery along each knot component of the depicted links. On the other hand, Theorem 1.1 implies that contact -surgeries along these links result in contact 3-manifolds with vanishing contact invariants. It will be shown in Examples 1.8 and 1.10 that contact -surgeries along Legendrian two-component links in Figures 1 and 2 yield overtwisted contact 3-manifolds. It is still unknown whether the contact 3-manifold obtained by contact -surgery along the Legendrian link in Figure 3 is overtwisted or tight.
More generally, all rationally slice knots satisfy [13, Theorem 1.10].
Recall that a knot is rationally slice if there exists an embedded disk in a rational homology -ball such that . Examples of rationally slice knots include strongly amphicheiral knots and Miyazaki knots, i.e., fibered, amphicheiral knots with irreducible Alexander polynomial . In particular, the figure-eight knot is rationally slice but not slice.
Let be a Legendrian link in . Suppose is a Legendrian unknot with , and is a meridional curve of . Then, by Theorem 1.1, contact -surgery along yields a contact structure on with vanishing contact invariant. Hence by the classification of tight contact structures on  and , the contact 3-manifold is overtwisted.
In the special case of Theorem 1.1 where is a Legendrian unknot with Thurston-Bennequin invariant , contact -surgery on along yields the unique (up to isotopy) tight contact structure on . Hence, in this case, we may interpret the theorem as a result of contact -surgery along a Legendrian knot in . More generally, we have the following corollary.
Suppose is a Legendrian knot in , the contact connected sum of copies of . If is not null-homologous, then contact -surgery on along yields a contact 3-manifold with vanishing contact invariant for any positive integer .
We also consider contact 3-manifolds obtained by contact -surgeries along Legendrian two-component links in whose two components have linking number zero. There are examples of such contact 3-manifolds whose contact invariants do not vanish.
Contact -surgery on along the Legendrian two-component unlink whose two components both have Thurston-Bennequin invariant yields the contact 3-manifold with nonvanishing contact invariant.
[20, Exercise 12.2.8(c)] provides examples of Legendrian two-component links in by removing one of the two Legendrian unknot components in [20, Figure 12.4] along which contact -surgeries yield contact 3-manifolds with nonvanishing contact invariants. One of the resulted contact surgery diagrams can be transformed to Figure 4 by Legendrian Reidemeister moves. One should be careful that there is a wrong crossing in [20, Figure 12.4].
Consider contact 3-manifolds obtained from by contact -surgeries along Legendrian Whitehead links. To the best of our knowledge, the contact invariants and tightness of such manifolds have not been explicitly given in the literature.
Contact -surgery on along a Legendrian Whitehead link yields a contact 3-manifold with vanishing contact invariant.
In light of the above vanishing results in Theorem 1.1 and Proposition 1.6, one may wonder whether the contact manifolds studied there are tight or not. Although we have not yet been fully successful in answering this question, we came up with the following sufficient condition for contact -surgeries yielding overtwisted contact 3-manifolds. Note that this theorem is irrelevant to the linking number of the two components of the Legendrian link along which we perform contact -surgery. It is inspired by the work of Baker and Onaran [1, Proposition 4.1.10].
Suppose there exists a front projection of a Legendrian two-component link in the standard contact 3-sphere that contains one of the configurations exhibited in Figure 5, then contact -surgery on along yields an overtwisted contact 3-manifold.
Suppose there exists a front projection of a Legendrian two-component link in the standard contact 3-sphere that contains one of the configurations exhibited in Figure 7, then contact -surgery on along yields an overtwisted contact 3-manifold.
In Figure 8, and are parts of and , respectively. Contact -surgery along the Legendrian link in the left of Figure 8 yields an overtwisted contact 3-manifold. This is because we can transform the Legendrian link in the left of Figure 8 to that in the right of Figure 8 which contains the configuration in Figure 7(a) through Legendrian Reidemeister moves. It follows that contact -surgery along the Legendrian link in Figure 2 yields an overtwisted contact 3-manifold.
The remainder of this paper is organized as follows. In Section 2, we review basic properties of the contact invariant. We also reformulate the statement of Golla concerning the conditions under which contact -surgery along a Legendrian knot yields a contact 3-manifold with nonvanishing contact invariant. In Section 3, we go through the construction of the link surgery formula of Manolescu and Ozsváth in the special case of two-components links. We elaborate on the page of an associated spectral sequence and identify the relevant maps in the differential with the well-known and in the knot surgery formula of Ozsváth and Szabó . In Section 4, we analyze the page and give a proof of Theorem 1.1 based on diagram chasing. This idea is partly inspired by the work of Hom and Lidman , and also constitutes the most novel part of our paper. Due to some technical issues, the above argument does not apply to the linking number 0 case, so in Section 5, we use a different machinery in Heegaard Floer homology, namely, the grading to prove Proposition 1.6. Finally in Section 6, we prove Theorem 1.7 and Corollary 1.9.
The authors would like to thank Ciprian Manolescu, Tye Lidman, Jen Hom, Faramarz Vafaee, Eugene Gorsky and John Etnyre for helpful discussions and suggestions. Part of this work was done while the second author was visiting The Chinese University of Hong Kong, and he would like to thank for its generous hospitality and support. The first author was partially supported by Grant No. 11371033 of the National Natural Science Foundation of China. The second author was partially supported by Grant No. 11471212 of the National Natural Science Foundation of China. The third author was partially supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 14301215).
2. Preliminaries on contact invariants
Let be an oriented Legendrian two-component link in . Suppose is the linking number of and . The result of contact -surgery along is denoted by , where
is the topological surgery framing matrix.
Let be the cobordism from to induced by the surgery, and be with reversed orientation. One has the contact invariant
where is the mirror of . We have a map
In , Golla investigated the contact invariant of a contact manifold given by contact surgery along a Legendrian knot in . In particular, by [9, Theorem 1.1], the contact 3-manifold obtained by contact -surgery along the Legendrian knot () in has nonvanishing contact invariant if and only if satisfies the following three conditions:
Hence, if either or does not satisfy one of these three conditions, then it follows readily from the functoriality (1.1) that the contact invariant must vanish as well.
There exists a two-component link such that the knot type of each component has a Legendrian representative satisfying the above three conditions, but the link type of this two-component link has no Legendrian representative with both two components satisfying the above three conditions simultaneously [5, Section 5.6].
3. Link surgery formula for two-component links
In this section, we recall the link surgery formula for two-component links developed in . The link surgery formula is a generalization of the knot surgery formula given by Ozsváth and Szabó . The idea is to compute the Heegaard Floer homology of a 3-manifold obtained by surgery along a knot in terms of the mapping cone of the knot Floer complex, and more generally surgery along a link in terms of the hyperbox of the link surgery complex. In addition, the cobordism map is realized as the induced map of the inclusion of complexes in the system of hyperboxes [17, Theorem 14.3].
We go over the construction for two-component links. Suppose is an oriented link with two components and , and the linking number of and is . Suppose the topological surgery framing matrix is
We will see the computation of .
Let The affine lattice over is defined by
The elements of correspond to Spin structures on relative to .
Similarly, let . The elements of correspond to Spin structures on relative to for . Furthermore, let and represent the component with the given and opposite orientation, respectively. If is or where is or for , we can define
where is the other component of . One caveat is that is independent of for .
Now we consider Spin structures on . Let be the (possibly degenerate) sublattice of generated by the two columns of . We denote the quotient of in by . For any , there is a standard way of associating an element . This gives an identification of the set with .
Fix . Manolescu and Ozsváth constructed a hyperbox of complexes which is a twisted gluing of the following four squares of chain complexes:
Here, , , , and are generalized Floer complexes that can be determined from a given Heegaard diagram of the link . There are also maps ’s between these generalized Floer complexes which count holomorphic polygons in the Heegaard diagram. We refer the reader to  for details. In Figure 9, we exhibit a more concrete representation of for which a square is drawn at the lattice point , and the generalized complexes (), , and are at the lower left, the upper left, the lower right and the upper right corner of the square, respectively. Note that while and stay at the original lattice point, and map to the complexes at the lattice points and ), respectively.
It is often convenient to study by introducing a filtration and consider the associated spectral sequence. Here, we define the filtration to be the number of components of if . Thus, the complex at the lower left corner of each square has filtration level 2; the complex at the lower right or the upper left corner of each square has filtration level 1; and the complex at the upper right corner of each square has filtration level 0. Since the largest difference in the filtration levels is 2, the th differential in the spectral sequence, , mush vanish if . By , the associated spectral sequence has
Let us explain the page of the surgery chain complex in greater detail. Figure 10 exhibits a typical example of an page associated to a 2-dimensional hyperbox . Observe that is the internal differential of each generalized Floer complex. Hence we have at the lower left corner of the square at the lattice point , which turns out to be isomorphic to of a large surgery along in certain Spin structure. Similarly, at the upper left corner and at the lower right corner of each square are isomorphic to of large surgeries along and in certain Spin structures, respectively; and at the upper right corner of each square is isomorphic to .
Next, we consider the differential . Note that consists of a collection of short edge maps and that stay at the original lattice point, and another collection of long edge maps and that shift the position by the vectors and , respectively. The most relevant maps for our purposes are the ones that map into the homology at the upper right corner of each square. Under the above identification of with Heegaard Floer homology of large integer surgeries, we can identify the short edge map initiated from the upper left corner as
the long edge map initiated from the upper left corner as
the short edge map initiated from the lower right corner as
and the long edge map initiated from the lower right corner as
where is a sufficiently large integer. In fact, and are the induced homomorphisms of and and are equivalent to the vertical and horizontal maps and defined in , respectively. The same thing holds for and .
4. Vanishing contact invariants
Proof of Theorem 1.1.
It follows from that . We claim that it suffices to consider the case where the Thurston-Bennequin invariant . Otherwise, must be strictly less than by the inequality ([27, Theorem 1]), thus violating the condition of (2.4). This then implies the triviality of the contact invariant by our discussion near the end of Section 2.
We first treat the case where is a Legendrian unknot. We try to determine the contact invariant . Note that is the unique generator of . Hence by (2.3), is the image of the generator under the cobordism map , so is equivalent to being a zero map.
We resort to [17, Theorem 14.3] to understand this map, which identifies with the induced map of the inclusion
In order to prove that vanishes, it suffices to show that for each , , the generator of at the upper right corner of the square at the lattice point , is a boundary in the page of the spectral sequence (or equivalently, trivial in the page).
For the subsequent argument, we will still refer to Figure 10 for a schematic picture of the page of the spectral sequence, although we should point out that at present the surgery is performed along the link , and correspond to and in Figure 10, respectively, and the topological surgery framing matrix is
Since is an unknot, the homology group that was identified with at the lower right corner of the square at the lattice point is 1-dimensional. We denote the generator of the homology group by . Clearly, we have:
(1) If , then is an isomorphism, and is the trivial map. So
(2) If , then is the trivial map, and is an isomorphism. So
(3) If , then both and are isomorphisms. So
On the other hand, we understand the general properties of the maps well when is sufficiently large. In that case, the homology group , and is an isomorphism while is the trivial map. Thus, if we denote the generator of the homology group at the upper left corner of the square at the lattice point by , then
Let us put them together. When , we can immediately see from Claims (1) and (2) that lies in the image of . When , we can use Claim (3) and (4.7) to find an explicit element such that under the assumption that the linking number is nonzero. More precisely, one can check that
for large enough and ; and
for large enough and . In either case, this proves that lies in the image of for each , thus implying the theorem for the special case where is a Legendrian unknot.
More generally, since satisfies , we apply (1.2) and conclude that is filtered chain homotopy equivalent to for some acyclic complex . Then, the above argument for the unknot case nearly extends verbatim to the general case, except that the homology group may not necessarily be 1-dimensional. Nevertheless, we can use the filtered chain homotopy equivalence to define the generators from the summand . The rest of the proof carries over for the general case. ∎
As a corollary, we show that contact -surgery on along a homologically essential Legendrian knot yields a contact 3-manifold with vanishing contact invariant for any positive integer , as claimed in Corollary 1.4.
Proof of Corollary 1.4.
The contact 3-manifold can be obtained by contact -surgery on along a Legendrian -component unlink . There exists a Legendrian knot in which becomes the Legendrian knot in after the contact -surgery along . To find such an , it suffices to perform Legendrian surgery on along a Legendrian -component link, each component of which lies in a summand and is disjoint from , so that the result is . Then the image of in is the desired .
Note that contact -surgery on along is equivalent to contact -surgery on along Legendrian push-offs of for any positive integer . Therefore, contact -surgery on along is equivalent to contact -surgery on along a Legendrian -component link , which is the union of the aforementioned Legendrian -component unlink and Legendrian push-offs of the Legendrian knot . See Figure 11 for an example.