All integral slopes can be Seifert fibered slopes\\for hyperbolic knots
Kimihiko Motegi
Hyun-Jong Song
Which slopes can or cannot appear as Seifert fibered slopes for hyperbolic knots in the $3$-sphere $S^3$? It is conjectured that if $r$-surgery on a hyperbolic knot in $S^3$ yields a Seifert fiber space, then $r$ is an integer. We show that for each integer $n \in \mathbb{Z}$, there exists a tunnel number one, hyperbolic knot $K_n$ in $S^3$ such that $n$-surgery on $K_n$ produces a small Seifert fiber space.
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%%%% All integral slopes can be Seifert fibered slopes for hyperbolic knots 
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%%%%             Kimihiko Motegi and Hyun-Jong Song
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%%%%          Published in Volume 5(2005) pages 369-378
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%%%%                Publication date 30 April 2005
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\documentclass{gtart_h}
\input agtout

\lognumber{16}
\volumenumber{5}
\volumeyear{2005}
\papernumber{16}
\pagenumbers{369}{378}
\received{10 March 2005} 
\revised{25 March 2005}
\accepted{12 April 2005}
\published{30 April 2005}


\usepackage{amsmath,amssymb,graphicx}

\newtheorem{LM}{Lemma}[section]
\newtheorem{THM}[LM]{Theorem}
\newtheorem{Q}[LM]{Question}
\newtheorem{PR}[LM]{Proposition}
\newtheorem{CO}[LM]{Corollary}
\newtheorem{CL}[LM]{Claim}
\newtheorem{CON}[LM]{Conjecture}
\newtheorem{AS}[LM]{Assertion}
\newtheorem{AD}[LM]{Addendum}
\theoremstyle{remark}
\newtheorem*{rem}{Remark}
\newtheorem*{rems}{Remarks}

\begin{document}

\title
{All integral slopes can be Seifert fibered slopes\\for hyperbolic knots} 

\author{Kimihiko Motegi\\Hyun-Jong Song}
\address{Department of Mathematics, Nihon University\\Tokyo 156-8550, Japan}
\secondaddress{Division of Mathematical Sciences, Pukyong National 
University\\599-1 Daeyondong, Namgu, Pusan 608-737, Korea}
\asciiaddress{Department of Mathematics, Nihon University\\Tokyo 
156-8550, Japan\\and\\Division of Mathematical Sciences, 
Pukyong National University\\599-1 Daeyondong, Namgu, Pusan 608-737, Korea}



\gtemail{\mailto{motegi@math.chs.nihon-u.ac.jp}{\rm\qua
and\qua}\mailto{hjsong@pknu.ac.kr}}
\asciiemail{motegi@math.chs.nihon-u.ac.jp, hjsong@pknu.ac.kr}




\begin{abstract}
	Which slopes can or cannot appear as Seifert fibered slopes 
for hyperbolic knots in the $3$-sphere $S^3$? 
It is conjectured that if $r$-surgery on a hyperbolic knot in 
$S^3$ yields a Seifert fiber space, 
then $r$ is an integer. 
	We show that for each integer $n \in \mathbb{Z}$, 
there exists a tunnel number one, 
hyperbolic knot $K_n$ in $S^3$ such that 
$n$-surgery on $K_n$ produces a small Seifert fiber space. 
\end{abstract}
\asciiabstract{%
Which slopes can or cannot appear as Seifert fibered slopes for
hyperbolic knots in the 3-sphere S^3?  It is conjectured that if
r-surgery on a hyperbolic knot in S^3 yields a Seifert fiber space,
then r is an integer.  We show that for each integer n, there exists a
tunnel number one, hyperbolic knot K_n in S^3 such that n-surgery on
K_n produces a small Seifert fiber space.}

\primaryclass{57M25, 57M50}
\keywords{Dehn surgery, hyperbolic knot, Seifert fiber space, surgery slopes}


\maketitle
{\small\it This paper is dedicated to Donald M. Davis on the occasion of his
60th birthday.\leftskip25pt\rightskip25pt\par}


\section{Introduction}
	Let $K$ be a knot in the $3$-sphere $S^3$ 
with a tubular neighborhood $N(K)$. 
Then the set of \textit{slopes} for $K$ 
(i.e., $\partial N(K)$-isotopy classes of simple loops on $\partial N(K)$) 
is identified with $\mathbb{Q} \cup \{\infty \}$ 
using preferred meridian-longitude pair so that 
a meridian corresponds to $\infty$. 
A slope $\gamma$ is said to be \textit{integral} if 
a representative of $\gamma$ intersects a meridian exactly once, 
in other words, $\gamma$ corresponds to an integer under the above 
identification. 
In the following, 
we denote by $(K; \gamma)$ 
the $3$-manifold obtained from $S^3$ by Dehn surgery
on a knot $K$ with slope $\gamma$, 
i.e., by attaching a solid torus to $S^3-$int$N(K)$ in such a way 
that $\gamma$ bounds a meridian disk of the filled solid torus. 
If $\gamma$ corresponds to $r \in \mathbb{Q} \cup \{ \infty \}$, 
then we identify $\gamma$ and $r$ and write $(K; r)$ for $(K; \gamma)$. \par

	We denote by $\mathcal{L}$ 
the \textit{set of lens slopes} 
$\{r \in \mathbb{Q}\ |\ \exists$ 
hyperbolic knot $K \subset S^3$ such that $(K;r)$ is a lens space$\}$, 
where $S^3$ and $S^2 \times S^1$ are also considered as lens spaces. 
Then the cyclic surgery theorem \cite{CGLS} implies that 
$\mathcal{L} \subset \mathbb{Z}$. 
A result of Gabai \cite[Corollary 8.3]{Ga} shows that 
$0 \not\in \mathcal{L}$, 
a result of Gordon and Luecke \cite{G-Lu} shows 
that $\pm 1 \not\in \mathcal{L}$. 
In \cite{KM} Kronheimer and Mrowka 
prove that $\pm 2 \not\in \mathcal{L}$. 
Furthermore, 
a result of Kronheimer, Mrowka, Ozsv\'ath and Szab\'o \cite{KMOS} 
implies that $\pm 3, \pm 4 \not\in \mathcal{L}$.  
Besides, 
Berge \cite[Table of Lens Spaces]{Berge} 
suggests that 
if $n \in \mathcal{L}$, 
then $|n| \ge 18$ and 
not every integer $n$ with $|n| \ge 18$ appears  
in $\mathcal{L}$. 
Fintushel and Stern \cite{FS} had shown that $18$-surgery on 
the $(-2, 3, 7)$ pretzel knot yields a lens space. 

\textit{Which slope $($rational number$)$ can or cannot appear in 
the set of Seifert fibered slopes 
$\mathcal{S} = \{r \in \mathbb{Q}\ |\ \exists$ 
hyperbolic knot $K \subset S^3$ such that $(K;r)$ 
is Seifert fibered$\}$?} 
It is conjectured that $\mathcal{S} \subset \mathbb{Z}$ \cite{Go}. 

The purpose of this paper is to prove: 

\begin{THM}
\label{Seifert slopes}
For each integer $n \in \mathbb{Z}$, 
there exists a tunnel number one, 
hyperbolic knot $K_n$ in $S^3$ 
such that $(K_n; n)$ is a small Seifert fiber space 
$($i.e., a Seifert fiber space over $S^2$ 
with exactly three exceptional fibers$)$. 
\end{THM}

\begin{rem}
Since $K_n$ has tunnel number one, 
it is embedded in a genus two Heegaard surface of $S^3$ and 
strongly invertible \cite[Lemma 5]{Mor}. 
See \cite[Question 3.1]{MMM}. 
\end{rem}

Theorem \ref{Seifert slopes}, 
together with the previous known results, 
shows: 

\begin{CO}
\label{LZS}
$\mathcal{L} \subsetneqq \mathbb{Z} \subset \mathcal{S}$. 
\end{CO}

\begin{rems}$\phantom{99}$

(1)\qua For the \textit{set of reducing slopes} 
$\mathcal{R} = \{r \in \mathbb{Q}\ |\ \exists$ 
hyperbolic knot $K \subset S^3$ such that $(K;r)$ is reducible$\}$, 
Gordon and Luecke \cite{G-Lu1} have shown that 
$\mathcal{R} \subset \mathbb{Z}$. 
In fact, the cabling conjecture \cite{GS} asserts that 
$\mathcal{R} = \emptyset$. 

(2)\qua For the \textit{set of toroidal slopes} 
$\mathcal{T} = \{r \in \mathbb{Q}\ |\ \exists$ 
hyperbolic knot $K \subset S^3$ such that $(K;r)$ is toroidal$\}$, 
Gordon and Luecke \cite{G-Lu2} have shown that 
$\mathcal{T} \subset \mathbb{Z}/2$ (integers or half integers). 
In \cite{Tera}, 
Teragaito shows that 
$\mathbb{Z} \subset \mathcal{T}$ and conjectures that 
$\mathcal{T} \subsetneqq \mathbb{Z}/2$. 
\end{rems}

\textbf{Acknowledgements}\qua
We would like to thank the referee for careful reading 
and useful comments. \newline
The first author was partially 
supported by Grant-in-Aid for 
Scientific Research (No.\ 15540095), 
The Ministry of Education, Culture, Sports, 
Science and Technology, Japan. \newline



\section{Hyperbolic knots with Seifert fibered surgeries}

	Our construction is based on an example of 
a longitudinal Seifert fibered surgery given in \cite{IMS}. 

Let $k \cup c$ be a $2$-bridge link given in 
Figure \ref{fig:knotKn}, 
and let $K_n$ be a knot obtained from $k$ by 
$\frac{1}{-n+4}$-surgery along $c$. 

\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.4\linewidth]{knotKn.eps}
\caption{$K_n$}
\label{fig:knotKn}
\end{center}
\end{figure}

        We shall say that a Seifert fiber space is of 
\textit{type} $S^2(n_1, n_2, n_3)$ 
if it has a Seifert fibration over 
$S^2$ with three exceptional fibers of 
indices $n_1, n_2$ and $n_3$ $(n_i \ge 2)$.  
Since $K_4$ is unknotted, 
$(K_4; 4)$ is a lens space $L(4, 1)$. 
For the other $n$'s, 
we have: 

\begin{LM}
\label{Seifert}
$(K_n; n)$ is a small Seifert fiber space of type 
$S^2(3, 5, |4n-15|)$ for any integer $n \ne 4$. 
\end{LM}

\begin{proof}
	Since the linking number of 
$k$ and $c$ is one (with suitable orientations), 
$(K_n ; n)$ has surgery descriptions 
as in Figure \ref{fig:description}. 

\begin{figure}[ht!]
\begin{center}
\includegraphics[width=1.0\linewidth]{description.eps}
\caption{Surgery descriptions of $(K_n; n)$}
\label{fig:description}
\end{center}
\end{figure}

	Let us take the quotient by the strong inversion of $S^3$ 
with an axis $L$ as shown in Figure \ref{fig:msequence1}. 



\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.9\linewidth]{msequence1.eps}
\nocolon\caption{}
\label{fig:msequence1}
\end{center}
\end{figure}

Then we obtain a branch knot $b'$ which is the image of the axis $L$. 
The Montesinos trick (\cite{Mon}, \cite{Bl}) shows that 
$-\frac{1}{2}, -1, \frac{3n-11}{-n+4}$ and 
$1$-surgery on 
$t_1, t_2, c$ and $k$ in the upstairs correspond to 
$-\frac{1}{2}, -1, \frac{3n-11}{-n+4}$ and 
$1$-untangle surgery on $b'$ in the downstairs, 
where an \textit{$r$-untangle surgery} is 
a replacement of $\frac{1}{0}$-untangle by 
$r$-untangle. 
(We adopt Bleiler's convention \cite{Ble} 
on the parametrization of rational tangles.) 
These untangle surgeries convert $b'$ into 
a link $b$ (Figure \ref{fig:msequence1}). 



\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.9\linewidth]{msequence2.eps}
\caption{Continued from Figure \ref{fig:msequence1}}
\label{fig:msequence2}
\end{center}
\end{figure}

	Following the sequence of isotopies in 
Figures \ref{fig:msequence1} and \ref{fig:msequence2}, 
we obtain a Montesinos link 
$M(\frac{2}{5}, -\frac{2}{3}, \frac{n-4}{4n-15})$. 

Since $(K_n ; n)$ is the double branched cover of $S^3$ branched over 
the Montesinos link $M(\frac{2}{5}, -\frac{2}{3}, \frac{n-4}{4n-15})$, 
$(K_n; n)$ is a Seifert fiber space of type 
$S^2(3, 5, |4n-15|)$ as desired. 
\end{proof}

\begin{LM}
\label{hyperbolic}
The knot $K_n$ is hyperbolic if 
$n \ne 3, 4, 5$. 
\end{LM}

\begin{proof}
Note that the $2$-bridge link given in Figure \ref{fig:knotKn} 
is not a $(2, p)$-torus link, 
and hence by \cite{Me} 
it is a hyperbolic link. 
If $n \ne 3, 4, 5$, 
then $|-n+4|>1$ and 
it follows from 
\cite[Theorem 1]{AMM1} (also \cite[Theorem 1.2]{AMM3})
that $K_n$ is a hyperbolic knot. 
See also 
\cite[Corollary A.2]{G-Lu3}, 
\cite[Theorem 1.2]{MM3} and 
\cite[Theorem 1.1]{AMM2}. 
\end{proof}

\begin{rem}
It follows from \cite{Ma}, \cite{KMS} 
that $K_n$ is a nontrivial knot except when $n = 4$. 
An experiment using Weeks' computer program 
``SnapPea" \cite{W} suggests that 
$K_3$ and $K_5$ are hyperbolic, 
but we will not use this experimental results. 
\end{rem}

\begin{LM}
\label{tunnel}
The knot $K_n$ has tunnel number one for any integer $n \ne 4$. 
\end{LM}

\begin{proof}
Since the link $k \cup c$ is a two-bridge link, 
the tunnel number of $k \cup c$ is one 
with unknotting tunnel $\tau$; 
A regular neighborhood $N(k \cup c \cup \tau)$ is a genus two 
handlebody and $S^3 - \mathrm{int}N(k \cup c \cup \tau)$ is also 
a genus two handlebody, 
see Figure \ref{fig:tunnel}. 

\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.5\linewidth]{tunnel.eps}
\nocolon\caption{}
\label{fig:tunnel}
\end{center}
\end{figure}

Then the general fact below 
(in which $k \cup c$ is not necessarily a two-bridge link) 
shows that the tunnel number of $K_n$ is less than or equal to one. 
Since our knot $K_n$ $(n \ne 4)$ is knotted in $S^3$, 
the tunnel number of $K_n$ is one.   
\end{proof}

\begin{CL}
\label{tunnel1}
Let $k \cup c$ be a two component link in $S^3$ 
which has tunnel number one. 
Assume that $c$ is unknotted in $S^3$. 
Then every knot obtained from $k$ by twisting along $c$ 
has tunnel number at most one. 
\end{CL}

\begin{proof}
	Let $\tau$ be an unknotting tunnel and 
$V$ a regular neighborhood of $k \cup c \cup \tau$ in $S^3$; 
$V$ is a genus two handlebody. 
Since $\tau$ is an unknotting tunnel for $k \cup c$, 
by definition, 
$W = S^3 - \mathrm{int}V$ is also 
a genus two handlebody. 
Take a small tubular neighborhood $N(c) \subset \mathrm{int}V$ 
and perform $-\frac{1}{n}$-surgery on $c$ using $N(c)$. 
Then we obtain 
a knot $k_n$ as the image of $k$ and 
obtain a genus two handlebody $V(c; -\frac{1}{n})$. 
Note that $V(c; -\frac{1}{n})$ and $W$ define 
a genus two Heegaard splitting of $S^3$, 
see Figure \ref{fig:arc}, 
where $c_n^*$ denotes the core of the filled solid torus. 

\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.88\linewidth]{arc.eps}
\nocolon\caption{}
\label{fig:arc}
\end{center}
\end{figure}

Then it is easy to see that an arc $\tau_n$ given by Figure \ref{fig:arc} 
is an unknotting tunnel for $k_n$ as desired. 
\end{proof}

	Now we are ready to prove Theorem \ref{Seifert slopes}. 
Lemmas \ref{Seifert}, \ref{hyperbolic} and \ref{tunnel} 
show that our knots $K_n$ enjoy the required properties, 
except for $n = 3, 4, 5$. 
To prove Theorem \ref{Seifert slopes}, 
we find hyperbolic knots $K'_n$ so that 
$(K'_n; n)$ is Seifert fibered for $n = 3, 4, 5$ 
(instead of showing that $K_3$, $K_5$ are hyperbolic). 
As the simplest way, 
let $K'_3$, $K'_4$ and $K'_5$ 
be the mirror image of 
$K_{-3}$, $K_{-4}$ and $K_{-5}$, 
respectively. 
Since $K_{-3}$, $K_{-4}$ and $K_{-5}$ are tunnel number one, 
hyperbolic knots 
by Lemmas \ref{hyperbolic} and \ref{tunnel}, 
their mirror images 
$K'_3$, $K'_4$ and $K'_5$ are also tunnel number one, 
hyperbolic knots. 
It is easy to observe that 
$(K'_3; 3)$ 
(resp. $(K'_4; 4)$, $(K'_5; 5)$) 
is the mirror image of $(K_{-3}; -3)$ 
(resp. $(K_{-4}; -4)$, $(K_{-5}; -5)$). 
By Lemma \ref{Seifert}, 
$(K_{-3}; -3)$, $(K_{-4}; -4)$ and $(K_{-5}; -5)$ are Seifert fibered, 
and hence $(K'_3; 3)$, $(K'_4; 4)$ and $(K'_5; 5)$ are also Seifert fibered. 
Putting $K_n$ as $K'_n$ for $n = 3, 4, 5$, 
we finish a proof of Theorem \ref{Seifert slopes}. 
\hspace*{\fill} $\qed$


\section{Identifying exceptional fibers}
	In \cite{MM3}, 
Miyazaki and Motegi conjectured that 
if $K$ admits a Seifert fibered surgery, 
then there is a trivial knot $c \subset S^3$ disjoint from $K$ 
which becomes a Seifert fiber in the resulting Seifert fiber space, 
and verified the conjecture for several Seifert fibered surgeries 
\cite[Section 6]{MM3}, see also \cite{EM}. 
Furthermore, 
computer experiments via ``SnapPea" \cite{W} suggest that 
such a knot $c$ is realized by a short closed geodesic in 
the hyperbolic manifold $S^3 - K$,
for details see \cite[Section 9]{MM3}, \cite{Mot2}. 

	In this section, 
we verify the conjecture for Seifert fibered surgeries 
given in Theorem \ref{Seifert slopes}. 

	Recall that $K_n$ is obtained from $k$ by 
$\frac{1}{-n+4}$-surgery on the trivial knot $c$ 
(i.e., $(n-4)$-twist along $c$), see Figure \ref{fig:knotKn}. 
Denote by $c_n$ the core of the filled solid torus. 
Then $K_n \cup c_n$ is a link in $S^3$ 
such that 
$c_n$ is a trivial knot. 

\begin{LM}
\label{fiber}
After $n$-surgery on $K_n$, 
$c_n$ becomes an exceptional fiber of index $|4n-15|$ 
in the resulting Seifert fiber space $(K_n; n)$. 
\end{LM}

\begin{proof}
	Following the sequences given by 
Figures \ref{fig:msequence1} and \ref{fig:msequence2}, 
we have a Montesinos link with three arcs
$\gamma$, $\tau_1$ and $\tau_2$ as in Figure \ref{fig:positions}, 
where $n = 1$ in the final Montesinos link, 
and $\gamma$, $\tau_1$, $\tau_2$ and $\kappa$ are the 
images of $c$, $t_1$, $t_2$ and $k$, respectively. 

\begin{figure}[ht!]
\begin{center}
\includegraphics[width=1.0\linewidth]{positions.eps}
\caption{Positions of exceptional fibers}
\label{fig:positions}
\end{center}
\end{figure}

From Figure \ref{fig:positions} we recognize that 
$t_1, t_2$ and $c$ become exceptional fibers of 
indices $5$, $3$ and $|4n-15|$, respectively 
in $(K_n; n)$. 
\end{proof}

	For $n \ne 3, 4, 5$, 
$c_n$ becomes an exceptional fiber of index $|4n-15|$, 
which is the unique maximal index,  
in $(K_n; n)$. 
	Experiments via ``SnapPea" \cite{W} 
suggest that $c_n$ is a shortest closed geodesic in $S^3 - K_n$ $(n \ne 3, 4, 5)$. 
For sufficiently large $|n|$, 
hyperbolic Dehn surgery theorem \cite{T1}, \cite{T2} 
shows that 
$c_n$ is the unique shortest closed geodesic in $S^3 - K_n$. \par

	Let us assume that $n = 3, 4, 5$. 
Then we have put $K_n$ as the mirror image of $K_{-n}$ in 
the proof of Theorem \ref{Seifert slopes}. 
Let $k' \cup c'$ be the mirror image of the link $k \cup c$. 
Then $K_n$ is obtained also from $k'$ by 
$\frac{1}{-n-4}$-surgery on $c'$ (i.e., $(n+4)$-twist along $c'$); 
we denote the core of the filled solid torus by $c'_n$.  
Note that there is an orientation reversing diffeomorphism 
from $(K_{-n}; -n)$ to $(K_n; n)$ sending 
$c_{-n}$ (regarded as a fiber in $(K_{-n}; -n)$) 
to $c'_n$ (regarded as a fiber in $(K_n; n)$). 
Thus the above observation implies that 
$c'_n$ becomes 
an exceptional fiber of index $|4n+15|$, 
which is the unique maximal index, 
in $(K_n; n)$ $(n = 3, 4, 5)$. 

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\bibitem{Mon} \textbf{J\,M Montesinos}; 
Surgery on links and double branched coverings of $S^3$, in:
Knots, groups, and $3$-manifolds (Papers dedicated to the memory of R\,H Fox), 
Ann.\ Math.\ Studies \textbf{84} (1975), 227--260.
  \MR{0380802}

\bibitem{Mor} \textbf{K Morimoto};
There are knots whose tunnel numbers go down under connected sum,
Proc. Amer. Math. Soc. \textbf{123} (1995), 3527--3532. 
  \MR{1317043}

\bibitem{Mot2} \textbf{K Motegi}; An experimental study of Seifert
fibered Dehn surgery via SnapPea, in: Proceedings of the Winter
Workshop of Topology/Workshop of Topology and Computer (Sendai,
2002/Nara, 2001), Interdisc. Inform. Sci. \textbf{9}
(2003), 95--125.  \MR{2023112}

\bibitem{Tera} \textbf{M Teragaito}; 
Toroidal surgeries on hyperbolic knots, 
Proc.\ Amer.\ Math.\ Soc.\ \textbf{130} (2002), 2803--2808. 
  \MR{1900888}

\bibitem{T1} \textbf{W\,P Thurston};
The geometry and topology of $3$-manifolds,
Lecture notes, Princeton University, 1979.\nl
\url{http://www.msri.org/publications/books/gt3m/}

\bibitem{T2} \textbf{W\,P Thurston};
Three dimensional manifolds, Kleinian groups and hyperbolic
geometry,
Bull.\ Amer.\ Math.\ Soc.\ 6 (1982), 357--381.
  \MR{0648524}

\bibitem{W} \textbf{J Weeks}; 
SnapPea: 
a computer program for creating and studying hyperbolic $3$-manifolds, 
freely available from 
\url{http://geometrygames.org/SnapPea/}

\end{thebibliography}
\Addresses\recd

\end{document}


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math0505322.extracted_bib
\begin{thebibliography}{99}

\bibitem{AMM1} \textbf{M A\"{\i}t Nouh}, \textbf{D Matignon}, \textbf{K Motegi}; 
Twisted unknots, 
C. R. Acad. Sci. Paris, Ser. I \textbf{337} (2003), 321--326. 
  \MR{2016983}

\bibitem{AMM2} \textbf{M A\"{\i}t Nouh}, \textbf{D Matignon}, \textbf{K Motegi}; 
Obtaining graph knots by twisting unknots, 
Topology Appl.\ \textbf{146--147} (2005), 105--121.
  \MR{2107139}

\bibitem{AMM3} \textbf{M A\"{\i}t Nouh}, \textbf{D Matignon}, \textbf{K Motegi};
Geometric types of twisted knots, preprint. 

\bibitem{Berge} \textbf{J Berge}; 
Some knots with surgeries yielding lens spaces, 
unpublished manuscript.

\bibitem{Ble} \textbf{S\,A Bleiler}; 
Knots prime on many strings, 
Trans.\ Amer.\ Math.\ Soc.\ \textbf{282} (1984), 385--401. 
  \MR{0728719}

\bibitem{Bl} \textbf{S\,A Bleiler}; 
Prime tangles and composite knots,
in: Knot theory and manifolds (Vancouver, B.C. 1983), 
Lect.\ Notes in Math.\ vol.\ \textbf{1144}, Springer-Verlag, 1985, 
pp. 1--13. 
  \MR{0823278}

\bibitem{CGLS} \textbf{M Culler}, \textbf{C\,McA Gordon}, 
\textbf{J Luecke}, \textbf{P\,B Shalen}; 
Dehn surgery on knots, 
Ann.\ Math.\  \textbf{125} (1987), 237--300.
  \MR{0881270}

\bibitem{EM} \textbf{M Eudave-Mu\~noz}; 
On hyperbolic knots with Seifert fibered Dehn surgeries, 
Topology Appl.\ \textbf{121} (2002), 
119--141. 
  \MR{1903687}

\bibitem{FS} \textbf{R Fintushel}, \textbf{R\,J Stern};
Constructing lens spaces by surgery on knots, 
Math.\ Z.\ \textbf{175} (1980), 33--51.
  \MR{0595630}

\bibitem{Ga} \textbf{D Gabai}; 
Foliations and the topology of 3-manifolds. III, 
J.\ Differential Geom.\ \textbf{26} (1987), 479--536.
  \MR{0910018}

\bibitem{GS} \textbf{F Gonz\'alez-Acu\~na}, \textbf{H Short}; 
Knot surgery and primeness, 
Math.\ Proc.\ Cambridge Philos.\ Soc.\ \textbf{99} (1986), 
89--102.
  \MR{0809502}

\bibitem{Go} \textbf{C\,McA Gordon}; 
Dehn Filling; a survey, 
Proc.\ Mini Semester in Knot Theory, 
Banach Center, Warsaw, Poland, 1995.

\bibitem{G-Lu1} \textbf{C\,McA Gordon}, \textbf{J Luecke}; 
Only integral surgeries can yield reducible manifolds, 
Math.\ Proc.\ Cambridge Philos.\ Soc.\ 
\textbf{102} (1987), 97--101. 
  \MR{0886439}

\bibitem{G-Lu} \textbf{C\,McA Gordon}, \textbf{J Luecke};
Knots are determined by their complements, 
J.\ Amer.\ Math.\ Soc.\ \textbf{2} (1989), 371--415.
  \MR{0965210}

\bibitem{G-Lu2} \textbf{C\,McA Gordon}, \textbf{J Luecke}; 
Dehn surgeries on knots creating essential tori, I,
Comm. Anal. Geom. \textbf{4} (1995), 597--644.
  \MR{1371211}

\bibitem{G-Lu3} \textbf{C\,McA Gordon}, \textbf{J Luecke}; 
Non-integral toroidal Dehn surgeries, 
Comm. Anal. Geom. \textbf{12} (2004), 417--485.
  \MR{2074884}

\bibitem{IMS} \textbf{K Ichihara}, \textbf{K Motegi}, \textbf{H-J Song}; 
Longitudinal Seifert fibered surgeries on hyperbolic knots, 
preprint. 

\bibitem{KMS} \textbf{M Kouno}, \textbf{K Motegi}, \textbf{T Shibuya};
Twisting and knot types,
J.\ Math.\ Soc.\ Japan 44 (1992), 199--216.
  \MR{1154840}

\bibitem{KM} \textbf{P Kronheimer}, \textbf{T Mrowka}; 
Dehn surgery, the fundamental group and SU(2). \arxiv{math.GT/0312322}

\bibitem{KMOS} \textbf{P Kronheimer}, \textbf{T Mrowka}, 
\textbf{P Ozsv\'ath}, \textbf{Z Szab\'o}; 
Monopoles and lens space surgeries. \arxiv{math.GT/0310164}

\bibitem{Ma} \textbf{Y Mathieu}, 
Unknotting, knotting by twists on disks and property (P) for
knots in $S^3$, Knots 90 (ed. by Kawauchi),
Proc. 1990 Osaka Conf. on Knot Theory and Related Topics,
de Gruyter, 1992, 93--102
  \MR{1177414}

\bibitem{MMM} \textbf{T Mattman}, \textbf{K Miyazaki}, \textbf{K Motegi}; 
Seifert fibered surgeries which do not arise from 
primitive/Seifert-fibered constructions, 
Trans.\ Amer.\ Math.\ Soc.\ (to appear). 

\bibitem{Me} \textbf{W Menasco}; 
Closed incompressible surfaces in alternating knot and link 
complements, 
Topology \textbf{23} (1984), 37--44.
  \MR{0721450}

\bibitem{MM3} \textbf{K Miyazaki}, \textbf{K Motegi}; 
Seifert fibered manifolds and Dehn surgery III, 
Comm.\ Anal.\ Geom.\ \textbf{7} (1999), 551--582. 
  \MR{1698388}

\bibitem{Mon} \textbf{J\,M Montesinos}; 
Surgery on links and double branched coverings of $S^3$, in:
Knots, groups, and $3$-manifolds (Papers dedicated to the memory of R\,H Fox), 
Ann.\ Math.\ Studies \textbf{84} (1975), 227--260.
  \MR{0380802}

\bibitem{Mor} \textbf{K Morimoto};
There are knots whose tunnel numbers go down under connected sum,
Proc. Amer. Math. Soc. \textbf{123} (1995), 3527--3532. 
  \MR{1317043}

\bibitem{Mot2} \textbf{K Motegi}; An experimental study of Seifert
fibered Dehn surgery via SnapPea, in: Proceedings of the Winter
Workshop of Topology/Workshop of Topology and Computer (Sendai,
2002/Nara, 2001), Interdisc. Inform. Sci. \textbf{9}
(2003), 95--125.  \MR{2023112}

\bibitem{Tera} \textbf{M Teragaito}; 
Toroidal surgeries on hyperbolic knots, 
Proc.\ Amer.\ Math.\ Soc.\ \textbf{130} (2002), 2803--2808. 
  \MR{1900888}

\bibitem{T1} \textbf{W\,P Thurston};
The geometry and topology of $3$-manifolds,
Lecture notes, Princeton University, 1979.\nl
\url{http://www.msri.org/publications/books/gt3m/}

\bibitem{T2} \textbf{W\,P Thurston};
Three dimensional manifolds, Kleinian groups and hyperbolic
geometry,
Bull.\ Amer.\ Math.\ Soc.\ 6 (1982), 357--381.
  \MR{0648524}

\bibitem{W} \textbf{J Weeks}; 
SnapPea: 
a computer program for creating and studying hyperbolic $3$-manifolds, 
freely available from 
\url{http://geometrygames.org/SnapPea/}

\end{thebibliography}