Compatible contact structures of fibered Seifert links in homology 3-spheres

Let $\Sigma$ be a homology $3$-sphere. We use the topological description of Seifert links in [5, p.60]. Let $\mathcal{S}=S^{2}\setminus\text{int}(D_{1}^{2}\cup\cdots\cup D_{k}^{2})$ be a 2-sphere with $k$ holes and make an oriented, closed, smooth $3$-manifold $\Sigma$ from $\mathcal{S}\times S^{1}$ by gluing solid tori $(D^{2}\times S^{1})_{1},\ldots,(D^{2}\times S^{1})_{k}$ along the boundary $\partial(\mathcal{S}\times S^{1})$. To fix the notation, we first choose a section $\mathcal{S}^{\text{sec}}$ of $\pi:\mathcal{S}\times S^{1}\to\mathcal{S}$ and set

Suppose that the gluing map of $(D^{2}\times S^{1})_{i}$ to $\mathcal{S}\times S^{1}$ is given so that $a_{i}Q_{i}+b_{i}H$ is null-homologous in $(D^{2}\times S^{1})_{i}$, where $(a_{i},b_{i})\in\mathbb{Z}^{2}\setminus\{(0,0)\}$ and $\gcd(|a_{i}|,|b_{i}|)=1$. To make the obtained $3$-manifold $\Sigma$ to be a homology $3$-sphere, the integers $a_{i}$’s and $b_{i}$’s should satisfy the equality $\sum_{i=1}^{k}b_{i}a_{1}\cdots a_{i-1}a_{i+1}\cdots a_{k}=\pm 1$. Following [5], in this paper, we always choose the coefficients $a_{i}$’s and $b_{i}$’s so that $\sum_{i=1}^{k}b_{i}a_{1}\cdots a_{i-1}a_{i+1}\cdots a_{k}=1$ by replacing $(a_{i},b_{i})$ into $(-a_{i},-b_{i})$ for some $i$ if necessary. Note that this equality ensures that if one of $a_{i}$’s is zero then all the other $a_{i}$’s satisfy $|a_{i}|=1$, and if $a_{i}\neq 0$ for all $i=1,\ldots,k$ then each pair $(i,j)$ with $i\neq j$ satisfies $\gcd(|a_{i}|,|a_{j}|)=1$. Since the $3$-manifold $\Sigma$ does not depend on the ambiguity of the choice of $b_{i}$’s, we may denote it as $\Sigma=\Sigma(a_{1},\ldots,a_{k})$.

The core curve $S_{i}$ of each solid torus $(D^{2}\times S^{1})_{i}$ is a fiber of the Seifert fibration after the gluings. We assign to $S_{i}$ an orientation in such a way that the linking number of $S_{i}$ and $a_{i}Q_{i}+b_{i}H$ equals $1$. This orientation is called the working orientation.

Let $(\mathfrak{m}_{i},\mathfrak{l}_{i})$ be the preferred meridian-longitude pair of the link complement $\Sigma\setminus S_{i}$ chosen such that the orientation of the longitude $\mathfrak{l}_{i}$ agrees with the working orientation of $S_{i}$. Then $(\mathfrak{m}_{i},\mathfrak{l}_{i})$ and $(Q_{i},H)$ are related by the following equations, see [5, Lemma 7.5]:

where $\sigma_{i}=a_{1}\cdots\hat{a}_{i}\cdots a_{k}$ and $\delta_{i}=\sum_{j\neq i}b_{j}a_{1}\cdots\hat{a}_{i}\cdots\hat{a}_{j}\cdots a_{k}$. Note that they satisfy $a_{i}\delta_{i}+b_{i}\sigma_{i}=1$.

Set $A=a_{1}\cdots a_{k}$. For a moment, we assume that $a_{i}\neq 0$ for all $i=1,\ldots,k$, in which case the orientation of the Seifert fibration in $\mathcal{S}\times S^{1}\to\mathcal{S}$ canonically extends into the fibers in $(D^{2}\times S^{1})_{i}$ for each $i=1,\ldots,k$, namely the orientation of the Seifert fibration of $\Sigma(a_{1},\ldots,a_{k})$ becomes well-defined. Note that the working orientation on $S_{i}$ coincides with the orientation of the Seifert fibration if and only if $a_{i}>0$.

We give the definition of fibered multilinks in $3$-manifolds. The same notion appears in [2], where the fibration is called a rational open book decomposition.

Let $M$ be an oriented, closed, smooth $3$-manifold and $L$ an unoriented link in $M$ with $n$ link components. We first assign an orientation to each link component of $L$, which we also call a working orientation. A multilink $L(\underline{m})$ in $M$ is a link each of whose components is equipped with a non-zero integer, called the multiplicity, where $\underline{m}=(m_{1},\ldots,m_{n})$ represents the set of multiplicities. A multilink $L(\underline{m})$ is called fibered if there is a fibration $M\setminus L\to S^{1}$ such that

• •

  the intersection of the fiber surface and a small tubular neighborhood $N(S_{i})$ of each link component $S_{i}$ of $L(\underline{m})$ locally consists of $|m_{i}|>0$ leaves meeting along $S_{i}$, and

• •

  the working orientation of $S_{i}$ is consistent with (resp. opposite to) the orientation induced from the fiber surface if $m_{i}>0$ (resp. $m_{i}<0$)

(cf. [5, p.28–29]).

A Seifert link $L$ in $\Sigma(a_{1},\ldots,a_{k})$ is a union of finite number of fibers of the Seifert fibration. We had introduced the working orientation for each link component $S_{i}$ of $L$ in Section 2.1. Using this working orientation, we assign a multiplicity to each $S_{i}$ and make $L$ to be a Seifert multilink. We denote this multilink as

where $1\leq n\leq k$. Note that Seifert multilinks are fibered in most cases and the fiberedness can be determined by a certain criterion stated in [5, Theorem 11.2]. A typical example of non-fibered Seifert multilink is the link obtained as the boundary of an $N$-times full-twisted annulus with $|N|\geq 2$.

Suppose that $L(\underline{m})$ is fibered. The interiors of the fiber surfaces of $L(\underline{m})$ intersect the fibers of the Seifert fibration transversely except for the case where $L(\underline{m})$ is a positive or negative Hopf multilink, see [5, Theorem 11.2] and the proof therein. In these exceptional cases, the transversality does not hold if the multiplicities and the denominators of the Seifert invariants satisfy a certain equation. As mentioned in [5, Proposition 7.3], a Seifert multilink is invertible and this involution changes $L(\underline{m})$ into $L(-\underline{m})$. In particular, this reverses the sign of the intersection of the interiors of the fiber surfaces and the fibers of the Seifert fibration. So, by choosing one of $L(\underline{m})$ and $L(-\underline{m})$ suitably, we often assume in this paper that the intersection is positive. We name it the positive transverse property and write it (PTP) for short.

Now we consider the case where $A=a_{1}\cdots a_{k}\neq 0$. In this case, as we already mentioned, the orientation of the Seifert fibration of $\Sigma(a_{1},\ldots,a_{k})$ becomes well-defined.

We first introduce terminologies in $3$-dimensional contact topology briefly, see for instance [22, 11] for general references.

A contact structure on $M$ is the $2$-plane field given by the kernel of a $1$-form $\alpha$ satisfying $\alpha\land d\alpha\neq 0$ everywhere on $M$. In this paper, we only consider a contact structure given by the kernel of a $1$-form $\alpha$ satisfying $\alpha\land d\alpha>0$, called a positive contact form on $M$. A vector field $R_{\alpha}$ on $M$ determined by the conditions $d\alpha(R_{\alpha},\cdot)\equiv 0$ and $\alpha(R_{\alpha})\equiv 1$ is called the Reeb vector field of $\alpha$. The $3$-manifold $M$ equipped with a contact structure $\xi$ is called a contact manifold and denoted as $(M,\xi)$. Two contact manifolds $(M_{1},\xi_{1})$ and $(M_{2},\xi_{2})$ are said to be contactomorphic if there exists a diffeomorphism $\varphi:M_{1}\to M_{2}$ such that $d\varphi:TM_{1}\to TM_{2}$ satisfies $d\varphi(\xi_{1})=\xi_{2}$. A disk $D$ in $(M,\xi)$ is called overtwisted if $D$ is tangent to $\xi$ at each point on $\partial D$. If $(M,\xi)$ has an overtwisted disk then we say that $\xi$ is overtwisted and otherwise that $\xi$ is tight. A typical example of overtwisted contact structures is given as follows: Let $\alpha$ be the contact form on $\mathbb{R}^{3}$ given by

where $(r,\theta,z)$ are coordinates of $\mathbb{R}^{3}$ with polar coordinates $(r,\theta)$. The contact structure $\ker\alpha$ is as shown in Figure 2. We can find an overtwisted disk in the tube $\{(r,\theta,z)\,;\,|r|\leq\pi+\varepsilon\}$, where $\varepsilon>0$ is a sufficiently small real number. Hence, this contact structure is overtwisted.

Now we introduce an effective way to describe a contact structure on $D^{2}\times S^{1}$. Let $\alpha$ be a $1$-form on $D^{2}\times S^{1}$ given by $\alpha=h_{2}d\mu-h_{1}d\lambda$, where $(r,\mu,\lambda)$ are coordinates of $D^{2}\times S^{1}$ with polar coordinates $(r,\mu)$ of $D^{2}$, and $h_{1}$ and $h_{2}$ are real-valued smooth functions with parameter $r$. We have

where $h_{1}^{\prime}$ and $h_{2}^{\prime}$ are the derivatives of $h_{1}$ and $h_{2}$ with parameter $r$ respectively. So, $\alpha$ is a positive contact form if and only if $h_{1}^{\prime}h_{2}-h_{1}h_{2}^{\prime}>0$. We now plot $(h_{1},h_{2})$ on the $xy$-plane. Since $(h_{2},-h_{1})$ represents a vector normal to the $2$-plane of the contact structure $\ker\alpha$, we can regard the line connecting $(0,0)$ and $(h_{1},h_{2})$ as the slope of $\ker\alpha$. The Reeb vector field $R_{\alpha}$ of $\alpha$ is given as

The parameter $r$ varies from $0$ to $1$, namely from $\{(0,0)\}\times S^{1}$ to the boundary of $D^{2}\times S^{1}$, and the pair of functions $(h_{1}(r),h_{2}(r))$ defines a curve $\gamma$ on the $xy$-plane. In summary, the curve $\gamma$ has the following properties:

• •

  Since $h_{1}^{\prime}h_{2}-h_{1}h_{2}^{\prime}>0$, $(0,0)\not\in\gamma([0,1])$ and $\gamma$ moves in clockwise orientation.

• •

  The line connecting $(0,0)$ and $(h_{1},h_{2})$ represents the slope of $\ker\alpha$ and the vector $(h_{2},-h_{1})$ represents the positive side of $\ker\alpha$.

• •

  The speed vector $(h_{1}^{\prime},h_{2}^{\prime})$ is parallel to $R_{\alpha}$ and points in the same direction.

See Figure 3. To make $\alpha$ to be a well-defined contact form in a neighborhood of $r=0$, we choose $\gamma$ near $r=0$ such that $(h_{1},h_{2})=(-c,r^{2})$ or $(h_{1},h_{2})=(c,-r^{2})$ with some positive constant $c$, so that $\alpha$ has the form $\alpha=r^{2}d\mu+cd\lambda$ or $\alpha=-(r^{2}d\mu+cd\lambda)$ near $r=0$ respectively.

If the curve $\gamma$ intersects the positive $x$-axis, then the contact structure $\ker\alpha$ on $D^{2}\times S^{1}$ has an overtwisted disk, similar to Figure 2, whose boundary corresponds to the intersection point of $\gamma$ and the positive $x$-axis. In this paper, we call the tube $(D^{2}\times S^{1},\ker\alpha)$ a Lutz tube and use it frequently to show the existence of an overtwisted disk.

The notion of compatible contact structures of fibered links can be generalized to fibered multilinks canonically. This idea also appears in [2]. Let $M$ be a closed, oriented, smooth $3$-manifold.

The next lemma gives a useful interpretation of the notion of compatible contact structures in terms of Reeb vector fields. In this paper we mainly use this characterization.

Now we introduce two fundamental facts concerning compatible contact structures of fibered multilinks, following the fibered link case.

Although the proof is analogous to the one in [31], since an explicit contact form of the compatible contact structure will be needed in the proof of Lemma 8.4 later, we prove the assertion here with presenting the contact form. A similar proof can be found in [2].

Throughout this section, we always assume that $A=a_{1}\cdots a_{k}>0$. Theorem 1.1 follows from the explicit construction of compatible contact structures described below.

To prove Proposition 4.1, we apply the argument in the proof in [31] to the Seifert fibration. We denote the boundary component $(-\partial\mathcal{S})\cap D^{2}_{i}$ of $\mathcal{S}$ by $C_{i}$.

We prepare two further lemmas which will be used for constructing the contact form on $(D^{2}\times S^{1})_{i}$. Let $B=[1,2)\times S^{1}\times S^{1}\subset\mathcal{S}\times S^{1}$ be a neighborhood of a boundary component of $\mathcal{S}\times S^{1}$ with coordinates $(r,\theta,t)$. We glue $D^{2}\times S^{1}$ to $B$ as

where $(\mathfrak{m},\mathfrak{l})$ is a standard meridian-longitude pair of $\partial D^{2}\times S^{1}\subset D^{2}\times S^{1}$, $Q$ is the oriented curve given by $\{1\}\times S^{1}\times\{\text{a point}\}\subset\partial B$, $H$ is a typical fiber of the projection $[1,2)\times S^{1}\times S^{1}\to[2,1)\times S^{1}$ which omits the third entry, and $a,b,\sigma,\delta\in\mathbb{Z}$ are given according to relations (2.1). The fibers $H=\sigma\mathfrak{m}+a\mathfrak{l}$ of the Seifert fibration on $\partial D^{2}\times S^{1}$ are canonically extended to the interior of $D^{2}\times S^{1}$.