Transverse doubled knots and clasp disks
Ichiro Torisu
We analyze transverse doubled knots in the standard contact 3-space by using spanned clasp disks. As applications, we will estimate their self-linking number and furthermore we will show that in many cases, transverse twist knots with the maximal self-linking number are unique up to transverse isotopy.
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\title{Transverse doubled knots and clasp disks }
\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 2, 2005.}





\maketitle



\abstract{We analyze transverse doubled knots in the standard
contact 3-space by using spanned clasp disks. As applications, 
we will estimate their self-linking number
and furthermore we will show that in many cases,  transverse twist knots
with the maximal  self-linking number are unique up to transverse
isotopy.}


\setlength\baselineskip{20pt}

\section {Introduction}
%\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} is an oriented
knot which is everywhere transverse  to the contact structure.
Throughout, we always assume a transverse knot is positively oriented  to
$\xi_0$. A {\it Legendrian knot} is an oriented knot
which  is everywhere tangent to the contact structure. The study of
transverse and Legendrian knots is very important in both 3-dimensional 
contact geometry and low-dimensional topology. For a reference, 
see J. Etnyre's survey article [9].


 
 In [4], Y. Eliashberg classified topologically trivial
transverse  knots in $(R^3,\xi_0)$ up to transverse isotopy in terms of
their self-linking number.  In [6], Eliashberg--M. Fraser classified
topologically trivial Legendrian  knots in $(R^3,\xi_0)$ up to Legendrian
isotopy in terms of their 
Thurston--Bennequin invariant and rotation number.
A topological knot type in $(R^3,\xi_0)$ is called  {\it
transversely simple}  ({\it Legendrian simple,} resp.) if
transverse (Legendrian, resp.) knots in this type are 
 determined by their self-linking number
(Thurston--Bennequin invariant and rotation number,  resp.).
Therefore, in other words, Eliashberg and Eliashberg--Fraser 
showed that the trivial knot  is transversely and Legendrian simple. 
Moreover recently, Etnyre and K. Honda proved  that 
 torus knots and the figure-eight knot are  both transversely and Legendrian 
 simple ([8], [10]). For Legendrian non-simple knot types, Y. Chekanov
and  Eliashberg--H. Hofer independently 
 discovered such examples ([3], [5], [7]). By computing a {\it contact
homology invariant}, they succeeded in distinguishing  a pair of
Legendrian knots whose topological knot type, Thurston--Bennequin
invariant and rotation number agree.
 Notice that their simplest topological knot type is a {\it twist knot} and a complete classification of a Legendrian 
 twist knot except Legendrian trefoil and figure-eight knots is still not done.   
 On the other hand, recently, J. Birman--W.
Menasco and  Etnyre--Honda, respectively, succeeded in finding
examples of transversely non-simple  knot type ([2], [11]). At writing
this paper, however, it is not known whether  a twist knot except the
trefoil and figure-eight knots is transversely simple.

In this paper, we geometrically analyze transverse doubled and twist knots in $(R^3,\xi_0)$
 by using spanned clasp disks. As applications, 
we will estimate their self-linking number  
and furthermore we will show that in many cases, transverse twist knots
with the maximal  self-linking number are unique up to transverse
isotopy.

\vspace{0.2cm}

\section{Main Results}	
\vspace*{-0.5pt}

First we recall classical invariants of transverse and Legendrian knots
in the standard contact 3-space $(R^3,\xi_0)$.
Let $v$ ($h$, resp.) be a non-singular vector field in $\xi_0$ 
(normal to $\xi_0$, resp.) on
$R^3$. 

\vspace*{5pt}
{\bf Definition~1.}
The {\it self-linking number} $sl(T)$ of a transverse knot $T$ 
in $(R^3,\xi_0)$ is 
the linking number of $T$ and a knot slightly pushed off $T$ along $v$.
\vspace*{5pt}

\vspace*{5pt}
{\bf Definition~2.}
The {\it Thurston--Bennequin invariant} $tb(L)$ of a Legendrian knot $L$ 
in $(R^3,\xi_0)$ is 
the linking number of $L$ and a knot slightly pushed off $L$ along 
$h$. The {\it rotation number} $r(L)$ is the rotation number of $L$ 
in the trivialization of $\xi_0$ given by $v$.
\vspace*{5pt}

Notice that $sl(T)$ ($tb(L)$, resp.) is independent of choices of $v$
 ($h$, resp.) and 
an orientation of $T$ ($L$, resp.) and that 
 $r(L)$ is independent of a choice of $v$, but 
$r(L)$ changes its sign if the orientation of $L$ is changed 
([1]). In [1], D. Bennequin showed that $sl(T)$ and
$tb(L)+|r(L)|$ are  bounded by its negative Euler
characteristic of Seifert surfaces  for $T$ and $L$,
respectively. 
%For example, if $T$ and $L$ are topologically 
%doubled knots, we obtain $sl(T)\leq 1$ and $tb(L)+|r(L)|\leq 1$. 

A topological knot $K$ in $R^3$ is a {\it doubled knot} of a companion knot
$K'$
  if $K$ is a satellite of $K'$ obtained by an image of 
the knot $J_+$ or $J_-$
  in the solid torus $D^2\times S^1$ featured in Figure
1,  where the tubular neighbourhood of $K'$ is replaced by the solid
torus.   A {\it twist knot} is a doubled knot whose   companion knot
is the trivial knot. A doubled knot is uniquely determined if 
 the framing $n$ of the image of $\{pt\}\times S^1, pt\in\partial D^2$ and
the knot 
$J_{\pm}$ are chosen ($n$ is an integer). 
In this paper, we call a doubled
knot (twist knot, resp.) with the chosen framing $n$ and $J_{+}$/$J_{-}$  a
$(J_{+}$/$J_{-},n)$-doubled knot ($(J_{+}$/$J_{-},n)$-twist knot, resp.).
For example, let $\rm{+3_1, -3_1, 
4_1}$ denote the positive trefoil, negative trefoil, 
figure-eight knots, respectively.  
Then $\rm{+3_1}$ is the 
$(J_{+},-1)$-twist knot, $\rm{-3_1}$ is the $(J_{-},1)$-twist knot, and
$\rm{4_1}$ is both the $(J_{+},1)$-twist knot and the $(J_{-},-1)$-twist
knot. 





\begin{figure}
\begin{center}
\includegraphics[width=10cm]{fig1.eps}
\end{center}
\caption{$(J_{\pm},n)$-twist knots}
\label{Figure 1}
\end{figure}%



For a topological knot type $K$, let $\bar{sl}(K)$ ($\bar{tb}(K)$, 
resp.) be the maximal value of the self-linking number 
(Thurston--Bennequin invariant, resp.) of all transverse 
(Legendrian, resp.) representative of $K$. For example, 
$\bar{sl}(\rm {O})$$=\bar{tb}(\rm {O})=-1$, 
$\bar{sl}(\rm{+3_1})$$=\bar{tb}(\rm{+3_1})=1$, 
$\bar{sl}(\rm{-3_1})=-5$, 
$\bar{tb}(\rm{-3_1})=-6$,
$\bar{sl}(\rm{4_1})$$=\bar{tb}(\rm{4_1})=-3$, where $\rm{O}$ denotes the
trivial knot (for example, see [10]). 



\vspace*{5pt}
{\bf Theorem~1.} Let $K$ be a $(J_{+},n)$-doubled knot of a
companion knot $K'$. Suppose  
$\bar{sl}(K)=2\bar{tb}(K')-2n+1$. 
Further suppose Legendrian knots in the knot type $K'$ with the maximal Thurston--Bennequin
invariant are unique up to Legendrian isotopy.   Then,  
transverse knots in the knot type $K$ with the maximal 
self-linking number are unique up to transverse isotopy.
\vspace*{5pt}

 
{\bf Corollary~1.} Let $K$ be a $(J_{+},n)$-twist knot for $n\geq -1$. 
Then, transverse knots in the knot type $K$ with the 
maximal self-linking number are unique up to transverse isotopy.
\vspace*{5pt}

{\bf Proof of Corollary 1.} 
By the Eliashberg's theorem in [4], topologically trivial transverse knots
in $(R^3,\xi_0)$ with  the maximal self-linking number $-1$ are unique. 
Therefore, by Theorem 1 
it is sufficient to show that $\bar{sl}(K)=-2n-1$. If $n\geq -1$, 
the lowest degree in the first variable of the HOMFLY polynomial 
of $K$ is easily calculated to be $-2n-1$. So we have 
$\bar{sl}(K)\leq -2n-1$ by the HOMFLY bound of self-linking 
number ([12], [14]). Finally, let $\bar{K}$ be a transverse 
$(J_{+},n)$-twist knot given by 
a  generic projection on $xz$-plane featured in  Figure 2. 
Note that the numbers of positive and negative 
crossings are 2 and $2n+3$, respectively and 
the writhe of $\bar{K}$ is $-2n-1$.  Therefore,    
it follows that $sl(\bar{K})=-2n-1$. Hence the inequality is sharp. 
\vspace*{5pt}

\begin{figure}
\begin{center}
\includegraphics[width=8cm]{fig2.eps}
\end{center}
\caption{A $xz$-projection of a transverse $(J_{+},n)$-twist knot}
\label{Figure 2}
\end{figure}%


\vspace*{5pt}
{\bf Remark~1.} 
The author does not know whether transverse knots in every twist 
knot type with the maximal self-linking number are unique 
up to transverse isotopy.
\vspace*{5pt}




\vspace*{5pt}
{\bf Theorem~2.}
Let $K$ be a $(J_{+},n)$-doubled knot of 
 a companion knot $K'$.
Then we have $\bar{sl}(K)\leq 2\bar{tb}(K')-2n+1$ and 
$\bar{tb}(K)\leq 2\bar{tb}(K')-2n+1$. 
\vspace*{5pt}


\vspace*{5pt}
{\bf Corollary~2.}
Let $K$ be a $(J_{+},n)$-twist knot. Then we have
$\bar{sl}(K)\leq -2n-1$ and 
$\bar{tb}(K)\leq -2n-1$.
\vspace*{5pt}

\vspace*{5pt}
{\bf Remark~2.} 
As mentioned in the proof of Corollary 1, if $n\geq -1$, the inequality
$\bar{sl}(K)\leq -2n-1$ of Corollary 2 is sharp and also deduced from the
HOMFLY bound. For a $(J_{+},n)$-twist knot with 
$n\leq -2$, this is not sharp in general. 
In [15], L. Ng proved that the Kauffman bound of the maximal 
Thurston--Bennequin invariant for a two-bridge knot is sharp. Since a
twist knot  is a two-bridge knot, 
the inequality
$\bar{tb}(K)\leq -2n-1$ of Corollary 2 
would be implied by that from the Kauffman bound.
\vspace*{5pt}


\section{Characteristic foliations on a clasp disk}	
\vspace*{-0.5pt}

Let $K$ be an oriented  doubled knot with companion $K'$ in $R^3$
and  let $\tilde{D}$ be a two-disk. 
Then there exists an immersion $f:\tilde{D}\rightarrow R^3$ 
such that $f_{|\partial\tilde{D}}:\partial\tilde{D}\rightarrow
K$ is a diffeomorphism and $f^{-1}(\Omega)$ consists 
of two arcs $\tilde\sigma_1, \tilde\sigma_2$ joining a point in
$\partial\tilde{D}$
 and a point of $f^{-1}(K\cap f(Int(\tilde D)))$ in
$Int\tilde D$, where 
$\Omega=\{x\in f(\tilde D)||f^{-1}(x)|\geq 2\}$. 
Put $D=f(\tilde D)$. 
 Throughout, we use the notation $\partial D$ as 
$f(\partial\tilde{D})$ (ie., $\partial D=K)$. Then 
$D$ is an oriented immersed disk in $R^3$
such that the induced orientation of
$\partial D$ agrees with the orientation of $K$.
 We call $D$ a
{\it clasp disk} for $K$ (Figure 3) and $\sigma=f(\tilde\sigma_1)
=f(\tilde\sigma_2)$ the {\it clasp singularity}.
Throughout this paper, for a  $(J_{+}$/$J_{-},n)$-doubled knot, 
we only consider a naturally spanned clasp disk via $J_{+}$/$J_{-}$, 
respectively.

\begin{figure}
\begin{center}
\includegraphics[width=10cm]{fig3.eps}
\end{center}
\caption{A clasp disk}
\label{Figure 3}
\end{figure}%


Further, suppose $K$ is a transverse knot 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=10cm]{fig4.eps}
\end{center}
\caption{A characteristic foliation}
\label{Figure 4}
\end{figure}%



Let $e_{+}$ ($e_{-}$, resp.) and $h_{+}$ ($h_{-}$, resp.) be the
number of positive (negative, resp.) elliptic and positive
(negative, resp.) hyperbolic points in $D_{\xi_0}$,
respectively.   Then from $D_{\xi_0}$, we can
compute the self-linking number  of $\partial D$ as follows.

\vspace*{5pt}
{\bf Proposition~1 ([1, p. 133; Theorem 7]).}
$sl(\partial D)$ is equal to
$\rho(D)-e_{+}+e_{-}+h_{+}-h_{-}$, where $\rho(D)$ is the
linking number of $\partial D$ and a knot obtained by shifting 
$\partial D$ along the normal to $D$ in $R^3$.
\vspace*{5pt}

\vspace*{5pt}
{\bf Remark~3.}
$\rho(D)$ is $\pm 2$ according to the sign of $J_{\pm}$.
\vspace*{5pt}



In the next section, we will prove the following.

\vspace*{5pt}
{\bf Theorem~3.}
For any transverse  $(J_{+},n)$-doubled knot $K$ in
$(R^3,\xi_0)$,  there is a generic
clasp  disk $D$ for $K$ 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}

The goal of this section is to prove the following.

\vspace*{5pt}
{\bf Theorem~4.}
Let $K$ be a transverse doubled knot in
$(R^3,\xi_0)$. 
 Suppose that the self-linking number of $K$ is maximal and that 
there is a generic clasp disk $D'$ for $K$ such that 
the clasp singularity $\sigma'$ of $D'$ is Legendrian and  
$\xi_0$ is transverse to 
$D'$ along  $\sigma'$. Then there exists a new generic clasp disk
$D$ for $K$ satisfying the following: 
\begin{itemize}
\item the clasp singularity $\sigma$ of $D$ is Legendrian and  
$\xi_0$ is transverse to 
$D$ along  $\sigma$,
\item $e_{-}=h_{+}=0$ in $D_{\xi_0}$,
\item there is a
Legendrian embedded circle
$C$ in
$D$ which contains $\sigma$ and  all
positive elliptic and negative hyperbolic point in
$D_{\xi_0}$ alternatively along $C-\sigma$.
\end{itemize}
\vspace*{5pt}


Let $K$ be a transverse doubled knot in
$(R^3,\xi_0)$ and $D$ be a generic clasp disk for $K$.
Suppose that the clasp singularity $\sigma$ of 
$D$ is Legendrian and $\xi_0$ is transverse to 
$D$ along  $\sigma$.
\vspace*{5pt}

First, as fully discussed in [4], 
we will modify $D$ relative to both $\partial D$ and $\sigma$ so
that $e_{-}=h_{+}=0$ in $D_{\xi_0}$.

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 [4], [13]).}
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 
$f_t:\tilde D\rightarrow R^3,t\in[0,1],$ 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 
$f_1(\tilde D)$ has no singularity in $U\cap f_1(\tilde D)$.
\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 [4], 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.

It follows that 
 after a  small isotopy of $D$ relative to both $\partial D$ and 
$\sigma$,
we can assume  that
there is a map $\alpha$ from an abstract finite tree $G$ 
 to $R^3$ 
satisfying:
\begin{itemize}
\item $e_{-}=h_{+}=0$ in $D_{\xi_0}$,
\item $\alpha(G)$ contains all singular point in $D_{\xi_0}$, 
\item $\alpha$ is injective,
\item the image of $\alpha$ of all vertex of $G$ equals 
the set of all positive elliptic point in $D_{\xi_0}$,
\item the restriction of $\alpha$ to each edge of $G$ is 
 Legendrian  and 
the each image contains exactly one
interior  singular point which is negative hyperbolic.  
\end{itemize}
See Figure 5. 

\begin{figure}
\begin{center}
\includegraphics[width=10cm]{fig5.eps}
\end{center}
\caption{}
\label{Figure 5}
\end{figure}%



Next we recall {\it stabilizations} and {\it bypasses} 
([8], [10]).



\vspace*{5pt}
{\bf Definition~3.}
A transverse knot $T$ is a {\it single stabilization} 
of a transverse knot $T'$ if 
$T=\beta\cup A, T'=\beta\cup A'$ and $A\cup A'$ cobounds a disk
with only one positive elliptic and one negative
 hyperbolic points and $sl(T)=sl(T')-2$ (Figure 6).
The disk bounded by $A\cup A'$ is called a {\it transverse bypass} and
$T'$ is  called a {\it single destabilization} of $T$. 
\vspace*{5pt}

\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig6.eps}
\end{center}
\caption{A transverse bypass}
\label{Figure 6}
\end{figure}%



Let $D_{\xi_0}$ be as above.

Now after doing single destabilizations on transverse bypasses on
$D_{\xi_0}$ disjoint from  the clasp singularity $\sigma$, if 
necessary, 
we have a generic clasp disk $D'$ such that 
after a small modification of $D'_{\xi_0}$,  
there is a Legendrian
embedded arc $\delta$ in
$D'$ which satisfies the following:
\begin{itemize}
\item $\delta$ contains all positive elliptic
and negative hyperbolic point alternatively along it in
$D'_{\xi_0}$,
\item $\delta$ is disjoint from the clasp singularity $\sigma'$
of $D'$,
\item $\partial\delta$ is two positive elliptic points in
$D'_{\xi_0}$. 
\end{itemize}
Further, by the genericity of $D'_{\xi_0}$, we can assume that 
$\sigma'$ is not included in an unstable trajectory from a
negative  hyperbolic point. See Figure 7.

Then finally, suppose that $K$ has the maximal self-linking
number. Since a single destabilization increases 
the self-linking number 
by two, it follows that in the above $D'_{\xi_0}$, 
$\delta$ can be extended by unstable trajectories from
$\partial\delta$ to $\sigma'$ so that the union of 
the extended $\delta$ and $\sigma'$   is a
simple closed curve in
$D'$. Because if otherwise, we can do more single
destabilization. 

After smoothing the corners of the simple
closed curve, we have a 
desired Legendrian embedded circle $C$ in
$D'$ as in the statement of Theorem 4.
This completes the proof of Theorem 4. 


\begin{figure}
\begin{center}
\includegraphics[width=10cm]{fig7.eps}
\end{center}
\caption{}
\label{Figure 7}
\end{figure}%

 




\vspace*{5pt}
{\bf Remark~4.}
In Theorem 4, if $K$ is a $(J_{\pm},n)$-doubled knot, 
$tb(C)=n-e_{+}+1=n-h_{-}$.
\vspace*{5pt}

\section{Proof of Theorem 3}	
\vspace*{-0.5pt}

For the proof of Theorem 3, we need a little knowledge of 
Legendrian knot theory  for the standard contact 3-space
$(R^3,\xi_0)$ and  convex surface theory as in
[10], [13] and [16]. 

Let $K$ be a transverse $(J_{+},n)$-doubled
knot. 

%Without loss of generality, 
%we assume $K$ is a $(J_+,n)$-doubled knot. Let $\tilde V$ be a
%solid torus as in Figure 1, which contains $J_+$.

First, take a {\it Legendrian push off} $L$ of K
such that $L$ is $C^0$-close to $K$ and $L$ is a closed leaf on
the boundary of a standard neighbourhood of $K$  ([4], [9]). 
We remark that   
a Legendrian isotopy of $L$ induces
a transverse isotopy  of $K$ ([4], [9]).
Let $E$ be a clasp disk naturally spanned by the Legendrian $(J_{+},n)$-doubled knot $L$ 
and let $c$ be a simple closed curve on $E$ such that $c$ contains the clasp singurality of $E$ 
and that the topological knot type of $c$ is the companion of $K$.

Then it is not hard to see that 
there are a topologically  standard solid torus $V$ embedded in
$(R^3, \xi_0)$ and meridian disks $d_1,d_2$ of $V$ which
satisfy the following (Figure 8): 
\begin{itemize}
\item $E\subset Int V$,
\item $E$ intersects $d_i$ as an arc $(i=1,2)$,
\item  $L$ intersects $d_i$ geometrically twice and 
algebraically zero $(i=1,2)$,
\item a 3-ball $B$ cut by $d_1$ and $d_2$ in $V$ contains
 a neighbourhood of the clasp singurality of $E$, 
\item the intersection of $c$ and $B$ is an unknotted arc properly embedded in $B$, 
\item $\partial V$ is a convex surface whose dividing set is
 two parallel simple closed curves and their slopes are a
negative integer,
\item $\partial d_i$ is Legendrian and $d_i$ is a convex disk 
whose dividing set is an  arc $(i=1,2)$.
\end{itemize}

\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig8.eps}
\end{center}
\caption{}
\label{Figure 8}
\end{figure}%




Then as discussed in [16], by a contact isotopy of $(R^3,\xi_0)$ 
we can regard
$V$ as a ``thin'' tubular neighbourhood of a Legendrian unknot
 in $(R^3,\xi_0)$.
Without loss of generality, we consider  $L$ is given by a 
{\it front} in $xz$-plane in $(R^3,\xi_0)$. So we can assume that
$B, d_i$ and $L_{|B}$  are corresponding to a square, its
vertical edge and a front as in Figure 9, 
respectively $(i=,1,2)$. 

By using Legendrian Reidemeister moves in the square, if
necessary,  we can find a Legendrian clasp part for $L$
featured in Figure 10. 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 10.

\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig9.eps}
\end{center}
\caption{A front}
\label{Figure 9}
\end{figure}%

\begin{figure}
\begin{center}
\includegraphics[width=10cm]{fig10.eps}
\end{center}
\caption{Legendrian clasps}
\label{Figure 10}
\end{figure}%




From $D_L$, we can easily make the desired clasp disk for
$K$. This completes the proof. 

\vspace*{5pt}
{\bf Remark~5.} 
For a transverse $(J_{-},n)$-doubled knot, Theorem 3 does not hold in
general.  For instance, suppose a transverse 
$(J_{-},0)$-twist knot $K$ with the maximal self-linking number  spans
such a clasp disk $D$. Then by Theorem 4, we can assume that 
$e_{-}=h_{+}=0$ and $e_{+}=h_{-}+1$ in $D_{\xi_0}$.
Therefore by Proposition 1, we have $sl(\partial D)=-1-2e_+$. 
But since $K=\partial D$ is topologically trivial and 
$sl(K)=-1$, it follows that $e_{+}=0$, a contradiction.

\vspace*{5pt}




\section{Proof of Theorem 1 and Theorem 2}	
\vspace*{-0.5pt}
We are ready to prove Theorem 1.
Let $K_1$ and $K_2$ be transverse $(J_{+},n)$-doubled knots. 
Suppose that $K_1$ is equivalent to $K_2$ as a topological knot $K$
and that both $K_1$ and $K_2$ 
have the maximal self-linking number. 
Without loss of generality, we assume both
$K_1$ and $K_2$ have 
the same companion knot $K'$. 
By Theorem 3 and Theorem 4, 
 there exist a generic clasp disk $D_i$ for $K_i$ and a Legendrian circle
$C_i$ in $D_i$ as in the statement of Theorem 4 $(i=1,2)$.
Remark that   
$D_{1\xi_0}$ is diffeomorphic to $D_{2\xi_0}$. We orient $C_1$ and
$C_2$  in the same way along $D_1$ and $D_2$.  
$C_1$ is equivalent to $C_2$ as 
the companion knot of $K_i$. 
Since $sl(K_1)=sl(K_2)$,  we have $tb(C_1)=tb(C_2)$ by
Proposition 1 and Remark 4.

Now suppose $\bar{sl}(K)=1-2n+2\bar{tb}(K')$. Then by Proposition 1,
Remark 3 and Remark 4,  we have $tb(C_i)=\bar{tb}(C_i)$ $(i=1,2)$.

Further suppose Legendrian knots in the knot type $K'$ with the maximal Thurston--Bennequin number are unique.    
Then, it follows that $C_1$ and $C_2$ are Legendrian isotopic. 
So there exists an orientation preserving diffeotopy  
$g_t:R^3\rightarrow R^3$ such that $g_0=$ id, $g_t$ is a
contactomorphism of $\xi_0$ for  each $t$, and
$g_{1*}(C_1)=C_2$  with its orientations.
Without loss of generality, we can further 
assume $g_{1}(\sigma_1)=\sigma_2$ , where $\sigma_1,\sigma_2$ 
are the clasp singularities of $D_1,D_2$. 
Then, using $g_t$ and $D_1,D_2$, it is not hard to see $K_1$ and
$K_2$ are transversely isotopic. This completes the proof of Theorem 1. 



\vspace*{5pt}
{\bf Proof of Theorem 2.} 
Let $K$ be a transverse $(J_{+},n)$-doubled knot with companion $K'$.
Suppose $sl(K)=\bar{sl}(K)$. Then by Theorem 3 and Theorem 4, 
there exist
a clasp disk $D$ for $K$ and a Legendrian circle $C$ in $D$ 
as in the statement of Theorem 4. 
By Proposition 1 and Remark 4,  we have $2tb(C)-2n=\bar{sl}(K)-1$. 
Since $C$ is topologically equivalent to $K'$,  
  it follows that
$\bar{sl}(K)\leq 2\bar{tb}(K')-2n+1$.  

And let $L$ be a Legendrian knot which is topologically equivalent to 
$K$. Suppose $tb(L)=\bar{tb}(K)$.  
Since $sl(T_{\pm}(L))=tb(L)\mp r(L)$, where $T_{+}(L),T_{-}(L)$ 
are positive and negative transverse push offs of $L$, 
we also have
$\bar{tb}(K)\leq 2\bar{tb}(K')-2n+1$. 
\vspace*{5pt}




 












 
\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]} Y. Chekanov,
\textit{Differential algebra of Legendrian links},
Invent. Math.,
\textbf{150} (2002) 441--483.

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

\bibitem{[5]} Y. Eliashberg,
\textit{Invariants in contact topology},
Proceedings of the ICM (Berlin, 1998), Doc. Math.
\textbf{2} (1998) 327--338.


\bibitem{[6]} Y. Eliashberg--M. Fraser,
\textit{Classification of topologically trivial Legendrian knots},
Geometry, topology, and dynamics (Montreal, PQ, 1995), CRM Proc. 
Lecture Notes, AMS,
\textbf{15} (1998) 17--51.

\bibitem{[9]} J. Epstein--D. Fuchs--M. Meyer,
\textit{Chekanov-Eliashberg invariants and transverse
approximations of Legendrian knots},
Pacific J. Math.,
\textbf{201} (2001) 89--106. 




\bibitem{[7]} J. Etnyre,
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Geomtery and Topology,
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\bibitem{[8]} J. Etnyre,
\textit{Legendrian and transversal knots},
to appear in the Handbook of Knot Theory. 

\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,
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preprint. 

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\textit{Invariants of Legendrian and transverse knots in the standard contact space},
Topology,
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\bibitem{[11]} E. Giroux,
\textit{Convexit\'e en topologie de contact},
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\bibitem{[11]} V. Goryunov--J. Hill,
\textit{A Bennequin number estimate for transverse knots},
`Singularity Theory. Proceedings of the Conference dedicated to C.T.C.Wall's 60th birthday', 
London Mathematical Society Lecture Notes Series,
\textbf{263} (1999) 265--280. 


\bibitem{[11]} L. Ng,
\textit{Maximal Thurston-Bennequin Number of Two-Bridge Links},
Algebraic and Geometric Topology,
\textbf{1} (2001) 427--434. 


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




\end{thebibliography}










\end{document}
math0505023.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]} Y. Chekanov,
\textit{Differential algebra of Legendrian links},
Invent. Math.,
\textbf{150} (2002) 441--483.

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

\bibitem{[5]} Y. Eliashberg,
\textit{Invariants in contact topology},
Proceedings of the ICM (Berlin, 1998), Doc. Math.
\textbf{2} (1998) 327--338.


\bibitem{[6]} Y. Eliashberg--M. Fraser,
\textit{Classification of topologically trivial Legendrian knots},
Geometry, topology, and dynamics (Montreal, PQ, 1995), CRM Proc. 
Lecture Notes, AMS,
\textbf{15} (1998) 17--51.

\bibitem{[9]} J. Epstein--D. Fuchs--M. Meyer,
\textit{Chekanov-Eliashberg invariants and transverse
approximations of Legendrian knots},
Pacific J. Math.,
\textbf{201} (2001) 89--106. 




\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. 

\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]} D. Fuchs--S. Tabachnikov,
\textit{Invariants of Legendrian and transverse knots in the standard contact space},
Topology,
\textbf{66} (1997) 1025--1053. 

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

\bibitem{[11]} V. Goryunov--J. Hill,
\textit{A Bennequin number estimate for transverse knots},
`Singularity Theory. Proceedings of the Conference dedicated to C.T.C.Wall's 60th birthday', 
London Mathematical Society Lecture Notes Series,
\textbf{263} (1999) 265--280. 


\bibitem{[11]} L. Ng,
\textit{Maximal Thurston-Bennequin Number of Two-Bridge Links},
Algebraic and Geometric Topology,
\textbf{1} (2001) 427--434. 


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




\end{thebibliography}