数理解析研究所講究録 1223 巻 2001 年 107-118

107

Exceptional surgeries and genera of knots

東京工業大学大学院情報理工学研究科 1
市原 一裕 (Kazuhiro Ichihara) 2

Graduate School of information Science and Engineering,Tokyo Institute of Technology 3 .

1I ntroduction

A3-manifold is said to be hyperbolic if it is homeomorphic to the quotientof the 3-dimensional hyperbolic space via atorsion free kleinian groupacting as isometries. We also say that aknot in a3-manifold is hyperbolic
if it has the hyperbolic complement.
ADehn surgery is one of the well-known operations producing anew
3-manifold from aprescribed one. When a3-manifold and aknot in it are
given, one can yield alot of new 3-manifolds by performing Dehn surgeries The well-known Hyperbolic Dehn Surgery Theorem due to Thurston[22] says that all but finitely many Dehn surgeries on ahyperbolic knot
give hyperbolic 3-manifolds. Also see [19] for adetailed proof.
In view of this result, aDehn surgery along ahyperbolic knot is called
exceptional if it yields anon-hyperbolic manifold. Alot of study to know
which surgeries are exceptional. Awell arranged survey was given in [7].
In particular, it was shown that on the number of exceptional surgeries,
there exists auniversal upper bound $[9, 15]$ .
1152-8552$2\mathrm{E}$-mail:$\ovalbox{\tt\small REJECT}\overline{5^{\mathrm{r}}\backslash },\mathrm{f}\mathrm{f}1$I.

$\ovalbox{\tt\small REJECT}\subset\cross\star \mathbb{H}1\rfloor 2-12-1$.titech. .jp-okayama, Meguro, Tokyo 152-8552$\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{a}\Phi \mathrm{i}\mathrm{s}$$\mathrm{s}2-12-10$$\mathrm{a}\mathrm{c}$108
One of the main subject of the study of exceptional surgeries is thoseon knots in the 3-sphere . It is well-known that aDehn surgery on
aknot inis characterized by the surgery slope, and such slopes areparameterized by. With respect to this coordinate, the rangeof exceptional surgery slopes is unbounded. Some specific examples were
given in [4, Section 5].
In this article, we will give some bounds on the range of exceptional
surgery slopes with respect to the coordinate above in terms of the generaof knots. In the sequel, let $K(r)$ be the closed 3-manifold obtained by a
Dehn surgery on ahyperbolic knot $K$ inalong aslopeanddenote the genus of $K$ . Our first theorem which is based on [10] is thefollowing.
$S^{3}$$S^{3}$$\mathbb{Q}\cup\{\infty\}$$S^{3}$$r\neq\infty$$g$

Theorem 1. If $|r|>3\cdot$, then $K(r)$ is an irreducible 3-manif0ld
with infinite and word-hyperbolic fundamental group.
$2^{7/4}g$

Remark that an approximate value ofis 10.09. It is known that
the Thurston’s Geometrization Conjecture would imply that irreducible
3-manifolds with infinite and word-hyperbolic fundamental group are
actually hyperbolic. We will briefly review on this fact in the end of
Section 3.
Next, we restrict knots to some special classes and give more sharperbounds. One class which we will consider is that of amphicheiral knots.
Remark that an approximate value of 1-2
is 0.29.
$3\cdot 2^{7/4}$

Theorem 2. If $K$ is amphicheiral and $|r|>3\cdot 2^{7/4}\{g-(1-2^{-1/2})\}$ ,then $K(r)$ is an irreducible 3-manifold with infinite and word-hyperbolic
fundamental group.
The next result is based on joint work with Makoto Ozawa [13]. In
the study of exceptional surgeries, afruitful method is to consider somesurfaces in aknot complement or asurgered manifold. In this article, we
consider closed essential (i.e., incompressible and not -parallel)surfaces
in aknot complement which admit an annulus connecting the surface
and the knot. Note that when the knot is hyperbolic such asurface$\partial$

109
corresponds to asurface subgroup contains accidental parabolics in the
knot group.
Theorem 3. Suppose that the complement of $K$ contains a closed essential surface admitting an annulus connecting the surface and K. Under
this assumption, $if|r|\geq 4g+1$ , then $K(r)$ is hyperbolic.
By virtue of the Thurston’s Uniformization Theorem [23] (see [18] fordetail), the proof of Theorem 3which we will give is purely topological.
Also will be used the results on such surfaces by Ozawa and the author[11], [12].

Concerning the surgeries yielding lens spaces, the following conjecture
was proposed by Goda and Teragaito in [6].
along
Conjecture 1.1. If a Dehn surgery on a hyperbolic knot inslope $r\neq\infty$ yields a lens space, then the knot is fibered and $2g+8\leq$$|r|\underline{<}4g-1$ , where
denotes the genus of the knot
$S^{3}$$a$$g$

They gave an upper bound $12g-7$ and proved that no such surgeries
can occur for genus one knots. Our theorems give some new bounds
which are sharper than theirs in certain cases. The conjecture above andtheir results is one of the motivations of our work.
The author would like to thank Katura Miyazaki for useful suggestions. He also thanks to Kimihiko Motegi, Masakazu Teragaito for helpfulcomments and Han Yoshida for letting him know the result of Adams[1].

2Preliminaries
The notations used throughout the article are as follows. For atopolog, $|X|$ and $\chi(X)$ denote the interior, the boundical space $X$ , Int(X),ary, the number of connected components and the Euler characteristic of$\partial X$

110, respectively. For a subset of $X$ ,denotes the exterior of
in $X$ , that is, the closure of, wheredenotes the regular$X$neighborhood of in .
A3-manif0ld $M$ is called irreducible if every 2-sphere embedded in$M$ bounds
a3-ball.
By a Dehn filling, we mean the operation of attaching solid tori to a3manifold with toral boundaries. ADehn surgery on alink in a3-manif0ld
means the following operation. Remove the open regular neighborhoodof the link and then perform aDehn filling. It is well-known that every
closed, orientable 3-manifold is obtained by aDehn surgery on alink in
the 3-sphere $S^{3}[16]$ .
We call the isotopy class of anon-trivial simple closed curve on atorus
a slope. When a generator system for the first homology of atorus isfixed, slopes on the torus are parameterized by. In the sequel, wealways fix the standard meridian-longitude system for the first homologyof the peripheral torus of aknot in $S^{3}[21]$ .
A Dehn surgery on aknot $K$ is determined by its surgery slope. Thatis, the slope of the meridian of the attached solid torus uniquely determines the homeomorphism type of the resultant 3-manif0ld.
The 3-manifold obtained from a3-manifold $M$ by Dehn filling along aslope is denoted by $M(r)$ . The 3-manifold obtained by aDehn surgery
on aknot $K$ inalong aslope is denoted by $K(r)$ .
$X$$\mathrm{Y}$$\mathrm{E}\mathrm{x}\mathrm{t}(\mathrm{Y})$$X-\mathrm{N}(\mathrm{Y})$$\mathrm{Y}$$\mathrm{N}(\mathrm{Y})$$\mathrm{Y}$$\mathbb{Q}\cup\{\infty\}$$r$$S^{3}$

3$r$

Proofs of Theorem 1 and Theorem 2

In this section, $M$ denotes a3-manifold with asingle toral boundary. Suppose that Int(M) admits a complete hyperbolic structure offinite volume. One can take ahoroball neighborhood of the cusp ofInt(M) and then identify
with the boundaryof $C$ . Sinceisregarded as aEuclidean torus as demonstrated in [22], the length of acurve oncan be defined. The length of aslope on
is definedas the minimum of the lengths of simple closed curves with slope , and$\partial M$$C$$\partial M$$\partial C$$\partial M$$r$$\partial C$$\partial M$

$r$

111
we denote it by $L(r)$ . Note that this length depends upon the choice of.
Let us prepare the following three lemmas. The next lemma wasshown by Agol [2], which was also obtained by Lackenby [17].
$C$

Lemma 3.1 ([2, Lemma 6.1]). If the length of a slope on
isgreater than 6, then the surgered manifold $M(r)$ is irreducible and its
fundamental group is infinite and word-hyperbolic .
$r$$\partial M$$\square$

One can take aparticular horoball neighborhood as follows. Takeamaximal one among those having no overlapping interior, and thenslightly shrink it. The next lemma holds for this , which was given in
$C$

$C$

[1].

Lemma 3.2 ([1, Theorem 5.3]). Every slope on
has the length
greater than, if $M$ is neither the figure-eight knot exterior, the exterior of the knot
in the knot table [21] nor the manifold obtained by-Dehn-filling on the Whitehead link exterior.
$\partial M$$2^{1/4}$$5_{2}$$(\mathit{2},\mathit{1})$$\square$

Aproperly immersed surface in $M$ is called essential if the immersion
induces injective maps of the fundamental groups and of the relative
fundamental groups. In [2], Agol proved the following.
Lemma 3.3 ([2, Lemma 5.1] ). Suppose that an essential surface
with boundary in $M$ is given. Let ,,be the slopes of boundary
components of S. Then $\sum_{i=1}^{n}L(r_{i})\leq 6|\chi(S)|$ .
$S$$r_{1}$$\ldots$$r_{n}$$\square$

Proof of Theorem 1. We first assume thatis the figure eight knot in. In this case, it is shown in [22] that if $K(r)$ is non-hyperbolic andthen $|r|\leq 4=4g$ .
Next, in the case that $K$ is the knot
in , it is also shown in [3]that if $K(r)$ is non-hyperbolic andthen $|r|\leq 4=4g$ .
Now, we consider ahyperbolic knot $K$ inneither the figure eight
knot nor the knot . Let $M$ denote the exterior of $K$ . Let $p/q$ be aslope on, where , are coprime integers and $q\neq 0$ . Suppose that$K$

$S^{3}$$r\neq\infty$$5_{2}$$S^{3}$$r\neq\infty$$S^{3}$$5_{2}$$\partial M$

$p$

$q$

112
$|p|>3\cdot 2^{7/4}g|q|$. By virtue of Lemma 3.1, we only need to show that$L(p/q)>6$ .

with
We choose ahoroball neighborhood as above and identify
in the universal cover. Let
be acomponent of the preimage of. By
gives alattice onof Int(M). The preimage of apoint on
fixing the base point , each primitive lattice point corresponds to a
, and the distance between and aprimitive lattice point isslope onequal to the length of the corresponding slope.
Take alattice point $P$ such that the path $OP$ is projected to themultiple of the longitude. We can take another primitive lattice point
corresponding to the slope $p/q$ such that the path $PQ$ is projected tomultiple of the meridian. Then, the triangle inequality gives that$C$$\partial C$$\partial M$$\overline{\partial C}$$\partial C$$\overline{\partial C}$$\partial C$

$O$

$O$

$\partial C$

$|q|$

$Q$

$|p|$

$|p|L(\infty)=PQ<OP+OQ=|q|L(0)+L(p/q)$

This implies that$L(p/q)>|p|L(\infty)-|q|L(0)$Let be the genus of $K$ , that is, the minimum of the genera ofSeifert surfaces for $K$ . Since aminimal genus Seifert surface is essential,$L(0)\leq 6(2g-1)$ holds by Lemma 3.3.
Combining this and Lemma 3.2, we conclude$g$$L(p/q)>3\cdot 2^{7/4}g|q|2^{1/4}-|q|6(2g-1)>6$ .
$\square$Aknot inmirror image.

$S^{3}$

is called amphicheiral if it is ambient isotopic to its

Proof of Theorem 2. In the same way as the proof above, we only needneither the figure eight knot nor
to consider ahyperbolic knot $K$ in
the knot 52 and will show that $L(p/q)>6$ . We use the same notations
be 3 $2^{7/4},1-2^{-1/2}$ , respectively
as the proof above and let ,$S^{3}$$C_{1}$$C_{2}$$\cdot$113
The key fact is that if $K$ is amphicheiral then the geodesies represented by meridian and the longitude are orthogonal in ahoroball neighborhood [20]. From this fact, the path $OP$ is orthogonal to $PQ$ , and so. Thus, one havethe angle $POQ$ is less than$\pi/2$$|p|^{2}L(\infty)^{2}=PQ^{2}>OP^{2}+OQ^{2}=|q|^{2}L(0)^{2}+L(p/q)^{2}$,

and$L(p/q)^{2}>|p|^{2}L(\infty)^{2}-|q|^{2}L(0)^{2}$

Together with the assumption that $|r|=|p/q|>C_{1}(g-C_{2})$ , $q\geq 1$
and the facts that $L(\infty)>2^{1/4}$ , $L(0)\leq 6(2g-1)$ , one has the next
inequalities.
$L(p/q)^{2}$$>$$\geq$$\{\sqrt{2}C_{1}^{2}(g-C_{2})^{2}-36(2g-1)^{2}\}|q|^{2}$$144(g-C_{2})^{2}-36(2g-1)^{2}$$=$$36(1-2C_{2})(4g-2C_{2}-1)$$=$$36(\sqrt{2}-1)(4g-3+\sqrt{2})$$\geq$

36

Consequently, we have that $L(p/q)>6$ .

$\square$

As we remarked in Section 1, the Thurston’s Geometrization Conjecture would imply that irreducible 3-manifolds with infinite and wordhyperbolic fundamental group are actually hyperbolic.
beHere, let us give adefinition of the word-hyperbolic group. Let
afinitely presented group. Fix afinite presentation of and let be the
Cayley graph of with respect to the presentation. One can regard as
ametric space by setting that each edge has length one. Then is called
word-hyperbolic if there exists apositive constant such that for every
geodesic triangle in , each one edge is contained in a neighborhood of
the other two edges. It can be proved that this definition does not dependon the choice of presentations [8]. It is shown that the fundamentalgroup of anegatively curved manifold is word-hyperbolic. Conversely,for 3-manifolds, the following is conjectured
$G$

$G$

$\Gamma$$G$$\Gamma$$G$$\delta$$\Gamma$$\delta$

114
Conjecture 3.1. If a closed, irreducible 3-manifold has infinite fundamental group which is word hyperbolic, then it is hyperbolic.

It is known that if aclosed, irreducible 3-manifold has infinite fundamental group which is word hyperbolic, then it is neither toroidal nor
Seifert fibered. A3-manifold is called toroidal if it contains an embedded essential torus, and it called Seifert fibered if it admits afoliation by
circles. Therefore, if the well-known Thurston’s Hyperbolization Conjecture, which says that such manifolds are actually hyperbolic, is affirmatively solved, then the consequence of our theorems is rewritten as that
$K(r)$ is hyperbolic.

4

Proof of Theorem 3

In this section, let $K$ be ahyperbolic knot in$S^{3}$$S^{3}$, $M$ the exterior of $K$ in.

Suppose that $M$ contains aclosed, essential, that is, incompressible
and not -parallel, embedded surface which admits an annulus connecting and $K$ .
This assumption is also stated in the following way. If $K$ is hyperbolic,$\pi_{1}(M)$ is identified withakleinian group , and if is essential, theinclusion map : $Sarrow M$ induces the monomorphism : $\pi_{1}(S)arrow G$ .
If thiscontains aparabolic element, then one can find an annulus
which runs from toby the annulus theorem. Then, by [5, Lemma
2.5.3], such an annulus determines either meridional or integral slope.
In the case that the slope is integral, such an annulus is regarded to
be running from to the knot $K$ . Moreover, in [12], such an annulus
is uniquely determined in that case up to isotopy. Consequently, theassumption above is equivalent to that there exist aclosed, essential
embedded surface such that
contains aparabolic element otherthan that represented by the meridian of $K$ .
To prove Theorem 3, we use the following lemma. By an annuluscompression, one obtains from an essential surface properly embedded$S$$\partial$

$S$

$G$

$i$

$S$

$i_{*}$$i_{*}(S)$$S$$\partial M$

$S$

$S$

$i_{*}(S)$

$S$

115
in M. We denote it by$S^{t}$. The boundary’$\partial S$

determines aslope a on

OM.
Lemma 4.1 ([13, Lemma 3.1] ). Let $F$ be an essential surface andthe slope determined by. Then,, wheredenotes the minimal geometric intersection number of the slopes.
$f$$\partial F$$\triangle(\alpha, f)\leq-2\chi(F)/|\partial F|$$\triangle(\alpha, f)$Here, we only describe the outline of its proof. See [13] for detail.
The key to prove this lemma is the fact that at least one componentofis essential in. By using this fact and the analysisof the graph appearing as $F\cap S’$ , one obtains an upper bound on thenumber of components of $F\cap S’$ .
$\partial \mathrm{E}\mathrm{x}\mathrm{t}(S)$$\mathrm{E}\mathrm{x}\mathrm{t}(5)$Proof of Theorem 3. By virtue of Thurston’s Uniformization Theorem[23], we only need to show that $K(r)$ is irreducible, remains essential
in $K(r)$ , $K(r)$ is not Seifert fibered and $K(r)$ contains no essential tori.
We set that $r=p/q$ and $\alpha=p’/q’$ , where and , and are c0prime integers, and assume that , $q’\neq 0$ . Then, their minimal geometricintersection number
are given as $|pq’-p’q|$ .
First, let us show that $K(r)$ is irreducible if $|r|=|p/q|\geq 4g$ . Suppose that $K(r)$ is reducible. Since $K$ is hyperbolic, the exterior $M$ isirreducible. Hence, the reducing sphere must intersect the attached solid
torus, and one can find an essential planer surface $F$ properly embedded
in $M$ . Note that the slope is represented by the boundary of $F$ . Let$m$ be the number of components of. Then, the next follows from
Lemma 4.1.
$S$

$p$

$q$

$p’$

$q’$

$q$

$\triangle(\alpha, r)$$r$$\partial F$.

$\triangle(\alpha, r)\leq\frac{-2(2-m)}{m}=2-\frac{4}{m}<2$

On the other hand, by considering aminimal genus Seifert surface, whoseboundary represents the slope 0, we have that$\triangle(\alpha, 0)\leq-2(2-2g-1)=4g-2$.

This implies that $|p’1-\mathrm{O}q’|=|p’|\leq 4g-2$ and that$\triangle(\alpha,r)=|pq’-p’q|\geq||p||q’|-|p’||q||\geq||p||q’|-(4g-2)|q||$116
Since the assumption is that $|p/q|\geq 4g$ , it follows that$\triangle(\alpha, r)\geq||p||q’|-(4g-2)|q||\geq|4g|q||q’|-(4g-2)|q||\geq|(4g|q’|-4g+2)|q||$Consequently, by , $|q’|>0$ , we conclude that
. This is acontradiction.
Next, we show that remains essential in $K(r)$ . This is an immediate
corollary of [5, Theorem 2.4.3]. It says that if, then remains
essential in $K(r)$ . As we showed above, in fact,holds.
Now, since $K$ is hyperbolic, the genus of is greater than one, andsince $K$ is aknot in , is separating in $K(r)$ . This implies that $K(r)$is not Seifert fibered [14, Theorem VI.34].
Finally, we show that $K(r)$ contains no essential tori if $|r|=|p/q|\geq$$4g+1$ . The argument to show this is almost same as that to show $K(r)$is irreducible. Suppose that $K(r)$ contains an essential torus. Since$K$ is hyperbolic, one can find an essential punctured torus $F$ properly
embedded in $M$ . Note that the slope is represented by the boundaryof $F$ . Let $m$ be the number of components of. Then, the next also
follows from Lemma 4.1.
$|q|$$\triangle(\alpha, r)\geq 2$$S$$\triangle(\alpha, r)\geq 2$$S$$\triangle(\alpha, r)\geq 2$$S$$S^{3}$$S$$r$$\partial F$$\triangle(\alpha, r)\leq\frac{-2(-m)}{m}=2$.

On the other hand, again by using that $|p’|\leq 4g-2$ , we have$\triangle(\alpha, r)=|pq’-p’q|\geq||p||q’|-|p’||q||\geq||p||q’|-(4g-2)|q||$

Since the assumption is that $|p/q|\geq 4g+1$ and$|q|$, $|q’|>0$ , it follows

that$\triangle(\alpha, r)\geq||p||q’|-(4g-2)|q||\geq|(4g+1)|q||q’|-(4g-2)|q||\geq 3$This is acontradiction, and completes the proof.

.
$\square$

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