On transverse Hopf links
Ichiro Torisu
We classify transverse Hopf links in the standard contact 3-space up to transverse isotopy in terms of their components' self-linking number.
Hopf.tex
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\begin{document}

\title{On transverse Hopf links}
\author{Ichiro Torisu}
\date{\footnotesize\it Naruto University of Education, 748,
Nakajima, Takashima, Naruto-cho, Naruto-shi, 772-8502 Japan\\ 
e-mail: torisu@naruto-u.ac.jp\\
Date: May 13, 2005.}





\maketitle



\abstract{We classify transverse Hopf links in the standard 
contact 3-space up to transverse isotopy in terms of their components'
self-linking number.}


\setlength\baselineskip{20pt}

\section {The statement of  result}
%\large

A {\it contact structure} on a 3-manifold is a completely non-integrable 
2-plane field on it.
Let $\xi_0$ be the standard contact structure on 3-space
$R^3=\{(x,y,z)\}$, that is,  a 2-plane field on $R^3$ defined by the
kernel of the 1-form ${dz-ydx}$.  A {\it transverse knot/link} is an oriented
knot/link which is everywhere transverse  to the contact structure.
Throughout, we always assume a transverse knot/link is positively oriented 
to $\xi_0$.

 Let $T$ be a transverse knot in 
the standard contact 3-space $(R^3,\xi_0)$ and 
 $v$ be a non-singular vector field in $\xi_0$  on $R^3$. Then
the {\it self-linking number} (aka. {\it Bennequin number}) 
$sl(T)$ of $T$ is defined to be
the linking number of $T$ and a knot slightly pushed off $T$ along $v$.
Note that $sl(T)$ is independent of choices of $v$ and 
an orientation of $T$,  $sl(T)$ is invariant under  
transverse isotopy of $T$ and that  $sl(T)$ equals the 
writhe in a generic projection of $T$ to $xz$-plane. 
In [1], D. Bennequin showed that $sl(T)$ is bounded by the negative Euler
characteristic of a Seifert surface for $T$.

In [3], Y. Eliashberg classified topologically trivial
transverse  knots in $(R^3,\xi_0)$ up to transverse isotopy in terms of
their self-linking number.  Therefore transverse knots featured in
Figure 1 give a complete list of transverse unknots in 
$(R^3,\xi_0)$, where the diagrams are drawn on $xz$-plane and the self-linking number 
is a negative odd integer.

\begin{figure}
\begin{center}
\includegraphics[width=7cm,height=4cm,keepaspectratio,clip]{fig1.eps}
\end{center}
\caption{A transverse unknot}
\label{Figure 1}
\end{figure}%


In [2], J. Birman--M. Wrinkle classified
topologically trivial transverse {\it links} in $(R^3,\xi_0)$ up to transverse
isotopy in terms of their components'  self-linking number.
Recently, J. Etnyre--K. Honda classified  transverse
 torus knots and transverse figure-eight knots in terms of
their self-linking number ([4], [6]).  But there 
is not much result of transverse links with more than one component.

In this paper, we will classify transverse {\it Hopf links} in the standard 
contact 3-space. For the proof, we will apply the Eliashberg's original arguments of analyzing 
a characteristic foliation of a disk to an immersed disk spanned by a Hopf link. 
Remark that in [8], the author also used the arguments for study of
transverse {\it doubled knots}.

Let $H_+=l_1^+\cup l_2^+$ and $H_-=l_1^-\cup l_2^-$ be the positive and negative Hopf links in $R^3$, where  $l_i^+$ and $l_i^-$ are the components and
the linking numbers $lk(l_1^+,l_2^+)$ and  $lk(l_1^-,l_2^-)$  are $-1$ and $1$, respectively.


\vspace*{5pt}
{\bf Theorem~1.}  Two transverse Hopf links in $(R^3,\xi_0)$ 
are transversely isotopic to each other 
if and only if their topological type $H_+/H_-$ and the pair of the self-linking number of 
their components agree.

\vspace*{5pt}

{\bf Remark~1.} (i) 
By Theorem 1, transverse links featured 
in Figure 2 give a complete
list of transverse Hopf links in $(R^3,\xi_0)$  up to transverse
isotopy, where the diagrams are drawn on $xz$-plane.\\
(ii) The statement of Theorem 1 implies that for a transverse Hopf
link in
$(R^3,\xi_0)$ with the same self-linking number of  two component,  
there is a transverse isotopy  which exchanges the
components.\\
(iii) After writing this paper, the author was 
pointed that the part for positive Hopf links is
proved in the Etnyre's positive torus knots paper
 [4] though he doesn't state that it was proved.

\vspace*{5pt}

\begin{figure}
\begin{center}
\includegraphics[width=9cm,height=8cm,keepaspectratio,clip]{fig2.eps}
\end{center}
\caption{transverse Hopf links}
\label{Figure 2}
\end{figure}%



\section{Characteristic foliations on immersed disks}	
\vspace*{-0.5pt}

Let $H=l_1\cup l_2$ be an oriented  Hopf link in $R^3$, where $l_i$ is the component.
Let $\tilde{D_1}\amalg \tilde{D_2}$ be a disjoint union of two 2-disks. 
Then there exists an immersion $f:\tilde{D_1}\amalg \tilde{D_2}\rightarrow R^3$ 
such that (i) $f_{|\partial\tilde{D_1}\amalg \partial\tilde{D_2}}:\partial\tilde{D_1}\amalg \partial\tilde{D_2}\rightarrow
H$ is a diffeomorphism and $f(\partial\tilde{D_i})=l_i$ (ii) $f^{-1}(\Omega)$ consists 
of a disjoint union  of two arcs $\tilde\sigma_1\amalg\tilde\sigma_2$ such that 
 each $\tilde\sigma_i$ joins a point of $\partial\tilde D_i$ to a point in
$Int\tilde D_i$, where  $\Omega=\{x\in f(\tilde{D_1}\amalg \tilde{D_2})||f^{-1}(x)|\geq 2\}$. 
Put $D=f(\tilde{D_1}\amalg \tilde{D_2})$, $D^1=f(\tilde{D_1})$ and $D^2=f(\tilde{D_2})$. 
Then $D=D^1\cup D^2$ is called  a {\it clasp disk} for $H$ 
and $\sigma=D^1\cap D^2=f(\tilde\sigma_1)
=f(\tilde\sigma_2)$ is called the {\it clasp singularity} (Figure 3). 
 Throughout, we use the notation $\partial D$ as 
$f(\partial\tilde{D_1}\amalg \partial\tilde{D_2})$ (ie., $\partial D=H)$. Then 
$D$ is an oriented immersed disk in $R^3$
such that the induced orientation of
$\partial D$ agrees with the orientation of $H$.



\begin{figure}
\begin{center}
\includegraphics[width=9cm,height=7cm,keepaspectratio,clip]{fig3.eps}
\end{center}
\caption{A clasp disk}
\label{Figure 3}
\end{figure}%


Further, suppose $H$ is a transverse link in $(R^3,\xi_0)$.
Let
$D_{\xi_0}$ be the {\it characteristic foliation} of $D$, that
is,  the singular foliation defined by $TD\cap \xi_0$ on $D$.
 Remark that $D_{\xi_0}$ is orientable.  
Throughout, we orient $D_{\xi_0}$ in such a way that
trajectories of $D_{\xi_0}$ exit through $\partial D$. 

  Generically $D_{\xi_0}$ satisfies the following: 
\begin{itemize}
\item $D$ is tangent to
$\xi_0$  at a finite set $\Sigma$ in $D-\partial D$ and
all singular point in $\Sigma$ is an {\it elliptic} point or a
{\it hyperbolic} point according to the index ($1$ or $-1$, 
respectively), 
\item  there is no
separatrix connection between hyperbolic points.
\end{itemize}

See Figure 4. In this paper, we call a  clasp disk satisfying
these 
 conditions a {\it generic} clasp disk. 
 
 And we
can distinguish between {\it positive} and {\it negative} 
points of $\Sigma$ depending on whether the orientations of
$D$  and $\xi_0$ coincide at these points or not.

\begin{figure}
\begin{center}
\includegraphics[width=8cm,height=7cm,keepaspectratio,clip]{fig4.eps}
\end{center}
\caption{A characteristic foliation}
\label{Figure 4}
\end{figure}%


Denote the restriction of $D_{\xi_0}$ to $D^i$ by $D_{\xi_0}^i$
$(i=1,2)$.  That is, $D_{\xi_0}^i$ is the characteristic foliation of
$D^i$. Let $e_{+}^i$ ($e_{-}^i$, resp.) and $h_{+}^i$ ($h_{-}^i$,
resp.) be the number of positive (negative, resp.) elliptic and
positive (negative, resp.) hyperbolic points in $D_{\xi_0}^i$,
respectively.   Then from $D_{\xi_0}^i$, we can
compute the self-linking number  of each component $l_i$ of $\partial D$ as follows.

\vspace*{5pt}
{\bf Proposition~1 ([1]).}
$sl(l_i)$ is equal to
$-e_{+}^i+e_{-}^i+h_{+}^i-h_{-}^i$ $(i=1,2)$.
\vspace*{5pt}

A curve in $(R^3,\xi_0)$ is called {\it Legendrian} if it is everywhere tangent to $\xi_0$.  

\vspace*{5pt}
{\bf Proposition~2.}
For any transverse  Hopf link $H$ in
$(R^3,\xi_0)$,  there is a generic
clasp  disk $D$ for $H$ such that 
the clasp singularity $\sigma$ of $D$ is Legendrian and  
$\xi_0$ is transverse to 
$D$ along  $\sigma$. In particular, there is 
 no singular point of the characteristic foliation
$D_{\xi_0}$ along $\sigma$. 
\vspace*{5pt}

{\bf Proof of Proposition 2.} 
Take a {\it Legendrian push off} $L$ of H
such that $L$ is $C^0$-close to $H$ and each component of $L$ is a closed leaf on
the boundary of a standard neighbourhood of each component of $H$  ([3], [5]). 
Note that a Legendrian isotopy of $L$ induces a transverse isotopy  of $H$ ([3], [5]).
Without loss of generality, we assume $L$ is given by a {\it front projection} on $xz$-plane in $R^3$. 
Then after Legendrian Reidemeister moves to $L$ , if
necessary,  we can find a Legendrian clasp part between two 
component of $L$ as in Figure 5. Then it is easy to see
that  there is 
 a clasp  disk $D_L$ for $L$ such that along the interior of
its clasp singularity $\sigma_L$, 
there is no tangency of $D_L$ to $\xi_0$, where $\sigma_L$ is
indicated by the dotted segment in Figure 5.
From $D_L$, we can easily make the desired clasp disk for
$H$. This completes the proof. 
\vspace*{5pt}

\begin{figure}
\begin{center}
\includegraphics[width=8cm,height=4cm,keepaspectratio,clip]{fig5.eps}
\end{center}
\caption{Legendrian clasps}
\label{Figure 5}
\end{figure}%

For the remainder of this section, we prove the following.

\vspace*{5pt}
{\bf Theorem~2.}
Let $H$ be a transverse Hopf link in $(R^3,\xi_0)$. 
  Then there exists a  generic clasp disk
$D=D^1\cup D^2$ for $H$ satisfying the following (Figure 6): 
\begin{itemize}
\item the clasp singularity $\sigma$ of $D$ is Legendrian and  
$\xi_0$ is transverse to 
$D$ along  $\sigma$,
\item $e_{-}^i=h_{+}^i=0$ in $D_{\xi_0}^i$ $(i=1,2)$,
\item $\sigma$ is included in an unstable trajectory from a positive elliptic point in $D_{\xi_0}^i$ $(i=1,2)$.
\end{itemize}
\vspace*{5pt}

\begin{figure}
\begin{center}
\includegraphics[width=12cm,height=10cm,keepaspectratio,clip]{fig6.eps}
\end{center}
\caption{}
\label{Figure 6}
\end{figure}%


{\bf Proof of Theorem 2.} 
By Proposition 2, there is a generic clasp disk $D$ for $H$ such that
 the clasp singularity $\sigma$ of $D$ is Legendrian and $\xi_0$ is transverse to 
$D$ along  $\sigma$.
\vspace*{5pt}

Then the next lemma due to E. Giroux is a basic tool of a modification of 
$D_{\xi_0}$ which 
allows us to cancel or create the pairs
of an  elliptic point and a hyperbolic point of the same sign in
$D_{\xi_0}$.

\vspace*{5pt}
{\bf Lemma~1 (Elimination lemma [3], [7]).}
Let $p, q\in D-\sigma$ be elliptic and hyperbolic points of
$D_{\xi_0}$
 of the same sign. Let $\gamma$ be a trajectory of $D_{\xi_0}$ 
 which joins $p$ and $q$ and is disjoint from $\sigma$. 
 Then there exists a $C^0$-small isotopy 
$f_t:\tilde{D_1}\amalg \tilde{D_2}\rightarrow R^3,t\in[0,1],$ which
starts with 
$f_0=f$, is fixed on $\gamma$ and outside of a neighbourhood 
$U\supset\gamma$ and such that the characteristic foliation of 
$f_1(\tilde{D_1}\amalg \tilde{D_2})$ has no singularity in $U\cap
f_1(\tilde{D_1}\amalg \tilde{D_2})$.
\vspace*{5pt}


We first
eliminate all negative elliptic point in $D_{\xi_0}$. 
Let $p$ be a negative elliptic point in $D_{\xi_0}$. If there is
a negative hyperbolic point connected with $p$ by a trajectory in
$D_{\xi_0}$, then we cancel the pair by Lemma 1.  If otherwise, 
as in the proof of Theorem 4.1.1 in [3], after creating the pairs 
of positive elliptic and positive hyperbolic points, if necessary,
we have  a limit cycle in $D_{\xi_0}$ which is disjoint from both 
$\partial D$ and $\sigma$. This contradicts the Bennequin
inequality. 

Next, we eliminate all positive hyperbolic point in $D_{\xi_0}$.
Let $q$ be a positive hyperbolic point in $D_{\xi_0}$. Then by
our assumptions, a stable separatrix  of $q$ necessarily connects
$q$ with a positive elliptic point. Again we cancel the pair by
Lemma 1. 

Finally, by the genericity of $D_{\xi_0}$, we can modify $D$ so that 
$\sigma$ is not included in an unstable trajectory from a
negative  hyperbolic point.  This completes the proof of Theroem 2.







\section{Proof of Theorem 1}


Recall from [4] that  a transverse link $K$ is a {\it single
stabilization}  of a transverse link $K'$ if 
$K=\beta\cup A, K'=\beta\cup A'$ and $A\cup A'$ cobounds a disk such that (i) the characteristic foliation of the disk 
admits only one positive elliptic and one negative hyperbolic points as singularities 
(ii) the interior of the disk is disjoint from $K'$ (iii) $sl(K)=sl(K')-2$ (Figure 7). 

\begin{figure}
\begin{center}
\includegraphics[width=7cm,height=5cm,keepaspectratio,clip]{fig7.eps}
\end{center}
\caption{}
\label{Figure 7}
\end{figure}%

For the proof of Theorem 1, we need the following Etnyre's theorem in [4]. 

\vspace*{5pt}
{\bf Theorem~3  ([4]).}  Let $(K,l)$, $(K',l')$, $(C,d)$ and $(C',d')$ be the pairs of 
transverse links and their one components. 
Suppose  transverse links $K$ and $K'$
are single stabilizations of transverse links $C$ and $C'$ and 
the stabilizations are done on the pairs $(l,d)$ and $(l',d')$, respectively.   
Then $(K,l)$ is  transversely isotopic to $(K',l')$ pairwisely if
 $(C,d)$ is transversely isotopic to $(C',d')$ pairwisely.
 \vspace*{5pt}
 
 In [4], Etnyre originally gave a proof for the case where $K,K',C$ and $C'$ are {\it knots}, 
 but the proof is directly applicable to our links case. 
 
By Theorem 2 and Theorem 3, we only need to prove  that  
transverse links in the link type $H_+/H_-$ such that the self-linking number of 
each component is maximal (i.e. $-1$) are unique up to transverse isotopy. 
In particular, for such a transverse Hopf link, there is a transverse
isotopy which exchanges  the components. 


The next lemma is easy to prove by considering front projections.
 \vspace*{5pt}
 
 {\bf Lemma~2.}  Suppose $c$ and $c'$ are oriented Legendrian
embedded 
 arcs in $(R^3,\xi_0)$. Then there is a contact isotopy of $(R^3,\xi_0)$ 
 which sends $c$ to $c'$ with their orientations.  
  \vspace*{5pt}


 
Let $H$ and $H'$ be such  transverse Hopf links. Then, 
 there exist generic clasp disks $D$ snd $D'$ for $H$ and $H'$
satisfying the statement of Theorem 2, respectively. 
Since the self-linking number of each component of $H$ and $H'$ 
 is maximal, the characteristic foliations $D_{\xi_0}$ and 
$D'_{\xi_0}$ admit
exactly two positive elliptic points as singularities, respectively.
 Let $c$ and $c'$ be 
 Legendrian arcs embedded in $D_{\xi_0}$ and $D'_{\xi_0}$ such that
the boundaries of $c$ and $c'$ equal the two positive elliptic
points of $D_{\xi_0}$ and $D'_{\xi_0}$  and that 
$c$ and $c'$ contain the clasp singularities of $D$ and $D'$,
respectively. 
 Then, for any choice of orientations  of $c,c'$, 
by Lemma 2 there is a contact isotopy of $(R^3,\xi_0)$
 which sends $c$ to $c'$ with the chosen orientations.
Furthermore, since $D_{\xi_0}$ is orientation preservings
diffeomorphic to $D'_{\xi_0}$,  by using $D$ and $D'$, 
 we can easily make a desired transverse isotopy from $H$ to $H'$. 
 This completes the proof of Theorem 1.  
 



 












 
\bibliographystyle{amsplain}
\begin{thebibliography}{10}

\bibitem{[1]} D. Bennequin,
\textit{Entrelacements et \'equations de Pfaff},
Ast\'erisque,
\textbf{107-108} (1983) 87--161.

%\bibitem{[2]} J. Birman--W. Menasco,
%\textit{Stabilization in the braid groups II: On transversal 
%simplicity of transversal knots},
%preprint. 

\bibitem{[3]} J. Birman--N. Wrinkle,
\textit{On transversally simple knots},
J. Differential Geometry,
\textbf{66} (2000) 325--354.


\bibitem{[4]} Y. Eliashberg,
\textit{Legendrian and transversal knots in tight contact 
3-manifolds},
Topological Methods in Modern Mathematics,
\textbf{} (1993) 171--193.





\bibitem{[7]} J. Etnyre,
\textit{Transversal torus knots},
Geomtery and Topology,
\textbf{3} (1999) 253--268. 

\bibitem{[8]} J. Etnyre,
\textit{Legendrian and transversal knots},
to appear in the Handbook of Knot Theory 
(available at http://front.math.ucdavis.edu/math.SG/0306256). 

\bibitem{[9]} J. Etnyre--K. Honda,
\textit{Knots and Contact geometry I: torus knots and the figure 
eight},
J. Symplectic Geometry,
\textbf{1} (2001) 63--120. 

%\bibitem{[10]} J. Etnyre--K. Honda,
%\textit{Cabling and transverse simplicity},
%preprint. 



\bibitem{[11]} E. Giroux,
\textit{Convexit\'e en topologie de contact},
Comment. Math. Helvetici,
\textbf{66} (1991) 637--677. 



%\bibitem{[12]} I. Torisu,
%\textit{On the additivity of the Thurston--Bennequin
%invariant  of Legendrian knots},
%Pacific J. Math.,
%\textbf{210} (2003) 359--365. 

\bibitem{[12]} I. Torisu,
\textit{Transverse doubled knots and clasp disks},
preprint (available at http://front.math.ucdavis.edu/math.GT/0505023).




\end{thebibliography}









\end{document}  
math0505297.extracted_bib
\begin{thebibliography}{10}

\bibitem{[1]} D. Bennequin,
\textit{Entrelacements et \'equations de Pfaff},
Ast\'erisque,
\textbf{107-108} (1983) 87--161.

%\bibitem{[2]} J. Birman--W. Menasco,
%\textit{Stabilization in the braid groups II: On transversal 
%simplicity of transversal knots},
%preprint. 

\bibitem{[3]} J. Birman--N. Wrinkle,
\textit{On transversally simple knots},
J. Differential Geometry,
\textbf{66} (2000) 325--354.


\bibitem{[4]} Y. Eliashberg,
\textit{Legendrian and transversal knots in tight contact 
3-manifolds},
Topological Methods in Modern Mathematics,
\textbf{} (1993) 171--193.





\bibitem{[7]} J. Etnyre,
\textit{Transversal torus knots},
Geomtery and Topology,
\textbf{3} (1999) 253--268. 

\bibitem{[8]} J. Etnyre,
\textit{Legendrian and transversal knots},
to appear in the Handbook of Knot Theory 
(available at http://front.math.ucdavis.edu/math.SG/0306256). 

\bibitem{[9]} J. Etnyre--K. Honda,
\textit{Knots and Contact geometry I: torus knots and the figure 
eight},
J. Symplectic Geometry,
\textbf{1} (2001) 63--120. 

%\bibitem{[10]} J. Etnyre--K. Honda,
%\textit{Cabling and transverse simplicity},
%preprint. 



\bibitem{[11]} E. Giroux,
\textit{Convexit\'e en topologie de contact},
Comment. Math. Helvetici,
\textbf{66} (1991) 637--677. 



%\bibitem{[12]} I. Torisu,
%\textit{On the additivity of the Thurston--Bennequin
%invariant  of Legendrian knots},
%Pacific J. Math.,
%\textbf{210} (2003) 359--365. 

\bibitem{[12]} I. Torisu,
\textit{Transverse doubled knots and clasp disks},
preprint (available at http://front.math.ucdavis.edu/math.GT/0505023).




\end{thebibliography}