# Transverse Surgery on Knots in Contact 3-Manifolds

###### Abstract.

We study the effect of transverse surgery on open books, the Heegaard Floer contact invariant, and tightness. We show that surgery on the connected binding of a genus open book that supports a tight contact structure preserves tightness if the surgery coefficient is greater than . We also give criteria for when positive contact surgery on Legendrian knots will result in an overtwisted manifold.

## 1. Introduction

On contact -manifolds, surgery on Legendrian knots is a well-storied affair, but much less well-studied is surgery on transverse knots. In this paper, we will systematically study transverse surgery, and its effect on open book decompositions, the Heegaard Floer contact invariant, and tightness. Along the way, we will see that many of the past results for surgery on Legendrian knots become more natural when phrased in the language of transverse surgery.

### 1.1. Admissible and Inadmissible Transverse Surgery

Ding and Geiges [8] show that any contact manifold can be obtained by contact surgery on a link in with its standard contact structure. Our first result combines with a previous result of Baldwin and Etnyre [6] to extend this statement to transverse surgery.

###### Theorem 1.1.

Every contact 3-manifold can be obtained by transverse surgery on some link. In particular, a single contact surgery (resp. surgery) on a Legendrian knot corresponds to admissible (resp. inadmissible) transverse surgery on a transverse knot.

In the other direction, Baldwin and Etnyre [6] showed that for a large range of slopes, admissible transverse surgery correspons to negative contact surgery on a Legendrian link in a neighbourhood of the transverse knot, but that for certain slopes, admissible transverse surgery was not comparable to any surgery on a Legendrian link. For inadmissible surgeries, we show:

###### Theorem 1.2.

Every inadmissible transverse surgery on a transverse knot corresponds to a positive contact surgery on a Legendrian approximation.

For contact surgeries on Legendrian knots where the surgery coefficient has numerator and denominator larger than 1, there is more than one choice of contact structure for the surgery. These correspond to ways to extend the contact structure over the surgery torus, and can be pinned down by choosing signs on bypass layers used to construct the surgery torus. We show that inadmissible transverse surgery corresponds to the contact structure obtained by choosing all negative bypass layers. Because of this, results that have been proved with some difficulty in the Legendrian setting are shown to be more naturally results about transverse knots and transverse surgery. As an example of this, we reprove a result of Lisca and Stipsicz [40] (in the integral case) and Golla [25] (in the rational case).

###### Corollary 1.3.

For any Legendrian knot , let be the contact structure obtained by choosing all negative signs on the bypasses for contact -surgery on . If is a negative stabilisation of , then is isotopic to .

### 1.2. Open Book Decompositions

Given any 3-manifold, there is an open book decomposition describing it. Here is a surface with boundary and is an orientation preserving diffeomorphism of fixing its boundary. The manifold is reconstructed from and as follows. The mapping torus of with monodromy is a 3-manifold with torus boundary components; a closed manifold is now obtained by gluing in solid tori to the boundary components such that a longitude (the binding) is mapped to the boundary components of the surfaces (the pages). We can define a contact structure on an open book decomposition that respects the decomposition, turning the binding into a transverse link. Open books support a unique contact structure up to isotopy, and thanks to work of Giroux [24], contact structures are supported by a unique open book up to a stabilisation operation.

Baker, Etnyre, and van Horn-Morris [2] demonstrated that rational open books, ie. those whose pages meet the binding in non-integral curves (or more than one boundary component of a page meet the same binding component), also support a unique contact structure up to isotopy. The curve that a page traces out on boundary of a neighbourhood of the binding is calling the page slope. Baker, Etnyre, and van Horn-Morris demonstrated that topological surgery on a binding component induces a rational open book, and if the surgery coefficient is less than the page slope, then the induced open book supports the contact structure obtained by admissible transverse surgery on that binding component. We look at surgery on a binding component with coefficient greater than the page slope and show the following.

###### Theorem 1.4.

The open book induced by surgery on a binding component with coefficient greater than the page slope supports the contact structure coming from inadmissible transverse surgery on the binding component.

Compare this with Hedden and Plamenevskaya [27], who discuss the same construction in the context of Heegaard Floer invariants. They track properties of the contact structure induced by surgery on the binding of an open book. We can now identify the contact structure they discuss as the one induced by inadmissible transverse surgery.

###### Construction 1.5.

We construct explicit integral open books that support admissible transverse surgery (less than the page slope) and inadmissible transverse surgery (with any surgery coefficient) on a binding component of an open book.

### 1.3. Tight Surgeries

The Heegaard Floer package provided by Ozsváth and Szabó [47, 46] gives very powerful 3-manifold invariants, which are still being mined for new information. Ozsváth and Szabó have shown [48] how a fibred knot gives rise to an invariant of the contact structure supported by the open book induced by the fibration. They proved that the non-vanishing of this invariant implies the contact structure is tight. Hedden and Plamenevskaya [27] showed that the same set-up and invariant exists when the knot is rationally fibred, ie. it is rationally null-homologous and fibred.

In light of Theorem 1.4, Hedden and Plamenevskaya proved that if is a fibred knot in supporting , and the contact class of is non-vanishing, then inadmissible transverse -surgery, for , where is the genus of , preserves the non-vanishing of the contact invariant. We extend this in two ways: first, we increase the range of the surgery coefficient to allow , and second, we replace the non-vanishing of the Heegaard Floer contact invariant with the weaker condition of tightness.

###### Theorem 1.6.

If is an integrally fibred transverse knot in supporting , where is tight (resp. has non-vanishing Heegaard Floer contact invariant), then inadmissible transverse -surgery for results in a tight contact manifold (resp. a contact manifold with non-vanishing Heegaard Floer contact invariant).

Given a knot , consider a product neighbourhood of , where . Let be the minimum genus of a surface in with boundary . When is a non-fibred transverse knot, and there is a Legendrian approximation of with , then we can conclude the following theorem. Since its proof is very similar to that of Theorem 1.6, we omit it; note that this result can also be extracted from the proof of a similar result of Lisca and Stipsicz for knots in [37].

###### Theorem 1.7.

If is a null-homologous transverse knot in with , where has non-vanishing contact class, and there exists a Legendrian approximation of with , then inadmissible transverse -surgery on preserves the non-vanishing of the contact class for .

To put these results in context, Lisca and Stipsicz [40] have a similar result to Theorem 1.7 for surgery on transverse knots , where . Mark and Tosun [43] generalise another result of Lisca and Stipsicz [37] to reduce that to , where . We do not recover their results entirely, as they do not require that there exist a Legendrian approximation with or respectively. In addition, Golla [25] can lower the surgery coefficient to for knots in with (along with requiring other technical conditions), where is the Heegaard Floer tau invariant; under a further technical condition, he can lower the coefficient to . However, we also discuss knots outside of , which the theorems of Lisca and Stipsicz, Mark and Tosun, and Golla do not. Hedden and Plamenevskaya’s [27] result does discuss knots outside of , and Theorem 1.6 is a generalisation of their result. In particular, Theorem 1.6 does not require non-vanishing of the contact invariant.

One might hope for an analogue of Theorem 1.6 for links, ie. that given an open book with multiple boundary components supporting a tight contact structure, sufficiently large inadmissible transverse surgery on every binding component might yield a tight manifold. We construct examples to show that no such theorem can exist. In particular, we construct the following examples.

###### Construction 1.8.

For every and , there is an open book of genus with binding a link with components supporting a Stein fillable contact structure, where anytime inadmissible transverse surgery is performed on all the binding components, the result is overtwisted.

### 1.4. Fillability and Universal Tightness

Seeing that tightness and non-vanishing of the Heegaard Floer contact invariant is preserved under inadmissible transverse surgery for large enough surgery coefficients, it is natural to ask whether other properties are preserved. Other properties that are preserved under contact -surgery are those of the various types of fillability: Stein, strong, or weak fillability. Results of Eliashberg [12], Weinstein [51], and Etnyre and Honda [18] show that contact -surgery preserves fillability in each of these categories.

Given a contact manifold , and a cover , there is an induced contact structure on . We say is universally tight if is tight and the induced contact structure on every cover is also tight; otherwise, we call the contact structure virtually overtwisted.

For large surgery coefficients, inadmissible transverse surgery adds a very small amount of twisting to the contact manifold. One might therefore expect that a sufficiently large surgery coefficient might preserve properties of the original contact structure, as in Theorem 1.6 and several other prior results. However, using classification results of Honda [29] and calculations of Lisca and Stipsicz [38], we construct examples of open books of all genera that show that inadmissible transverse surgery does not in general preserve the property of being universally tight, nor any of the fillability categories, even with arbitrarily large surgery coefficient.

###### Construction 1.9.

For every , there is a transverse knot of genus in a Stein fillable universally tight contact manifold, where sufficiently large inadmissible transverse surgery preserves the non-vanishing of the contact class, but where no inadmissible transverse surgery is universally tight or weakly semi-fillable.

### 1.5. Overtwisted Surgeries

Lisca and Stipsicz [39] show that given a Legendrian with , contact -surgery on has vanishing Heegaard Floer contact invariant. It is natural to ask whether these are overtwisted. We answer this question for a large class of knots, not just in , leaving out only a finite set of pairs for each knot genus . A sample result of this type is as follows.

###### Theorem 1.10.

Let be a null-homologous Legendrian in , where is torsion, with and , and let be a positive transverse push-off of . Then inadmissible transverse -surgery on is overtwisted. If , then all inadmissible transverse surgeries on are overtwisted.

Using this and its generalisation to contact -surgery, we derive the following corollary.

###### Corollary 1.11.

For every genus and every positive integer , there is a negative integer such that if is a null-homologous Legendrian knot of genus and , then contact -surgery on is overtwisted.

Lisca and Stipsicz [39] show that all contact -surgeries on negative torus knots are overtwisted. Our results allow for the following generalisation.

###### Corollary 1.12.

All inadmissible transverse surgeries on negative torus knots in are overtwisted.

Note that using the above results, we cannot show that all inadmissible transverse surgeries on the Figure Eight knot in are overtwisted. However, we can obtain this result using convex surface theory methods, see [7]. Note that inadmissible transverse -surgery on is equivalent to contact -surgery on .

### 1.6. Knot Invariants

Given a knot type in a contact manifold , we can consider all regular neighbourhoods of transverse representatives of . We define the contact width of be the supremum of the slopes of the characteristic foliation on the boundary of these neighbourhoods. The following follows from Theorem 1.6.

###### Corollary 1.13.

Let be a fibred transverse knot in a tight manifold with such that the fibration supports the contact structure . If the maximum Thurston–Bennequin number of a Legendrian approximation of is , then .

We also extend an invariant of transverse knots defined by Baldwin and Etnyre [6], and determine its value in certain cases.

### 1.7. Organisation of Paper

Section 2 contains a description of Legendrian and Transverse surgery, and proofs of Theorem 1.1. Section 3 describes open book decompositions and, proofs of Theorem 1.2, Corollary 1.3, Theorem 1.4, and the construction of open books supporting transverse surgery on a binding component (Construction 1.5). Section 4 discusses Heegaard Floer theory, proves Theorem 1.6 and Corollary 1.13, and discusses Construction 1.8. Section 5 discusses fillability, universal tightness, and Construction 1.9. Finally, Section 6 discusses when the result of surgery can be proved to be overtwisted, and proves Theorem 1.10.

### 1.8. Acknowledgements

The author would like to thank John Etnyre for his support and many helpful discussions throughout this project. He would like to thank Kenneth Baker for helpful discussions that led to the results in Section 6. The author would further like to thank Bülent Tosun and David Shea Vela-Vick, who made helpful comments on early drafts of this paper. This work was partially supported by NSF Grant DMS-13909073.

## 2. Contact and Transverse Surgery

In this section, we will give a background to the contact geometric concepts used throughout the paper. We will then describe contact and transverse surgery, and see that all contact surgeries can be re-written in terms of transverse surgeries.

### 2.1. Background

We begin with a brief reminder of standard theorems about contact structures on -manifolds which we will use throughout this paper. Further details can be found in [16, 13].

#### 2.1.1. Farey Tessellation

The Farey tessellation is given by a tiling of the hyperbolic plane by geodesic triangles shown in Figure 1, where the endpoints of the geodesics are labeled. Our convention is to use cardinal directions (specifically, North, East, South, and West) to denote points on the circle. These names will always refer to their respective locations, even as the labeling will move around. There is a standard labeling, shown in Figure 1, which is as follows: let West be labeled and East be labeled . The third unlabeled point of a geodesic triangle with two corners already labeled and is given the label . We denote this Farey sum operation . In the Southern hemisphere, by treating as , labeling the corners of geodesic triangles using the Farey sum suffices to label each endpoint of every geodesic with a positive number. Thus South gets labeled , and so on. In the Northern hemisphere, we label with negative numbers by treating as . Thus North gets labeled , and so on. Every rational number and infinity is found exactly once as a label on the Farey tessellation.

###### Remark 2.1.

For notation purposes, we denote the result of the Farey sum of and by . We also denote , where there are copies of , as . In general, we use the operation with two fractions even when they are not adjacent on the Farey tessellation. While in this case, there is no direct relation to operations on the Farey tessellation, we note that if points labeled and are connected by a geodesic in the tessellation, then is the label of some point in between those labeled and .

Other labelings of the Farey tessellation can be created by labeling any two points connected by a geodesic with 0 and . We can then construct a labeling by an analogous process to the above. In fact, this labeling is the result of a diffeomorphism applied to the hyperbolic plane with the standard labeling, but we will not need that fact explicitly here.

###### Remark 2.2.

If we consider each label as a reduced fraction in the obvious way (where the denominator of a negative number is negative), then each fraction naturally corresponds to a vector

The vectors corresponding to and , with which we started our labeling, constitute a basis for . Hence the vectors corresponding to the endpoints of any geodesic in the tessellation form a basis for .

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