International Journal of Mathematics and Mathematical Sciences Volume 2010, Article ID 573403,8pages
doi:10.1155/2010/573403
Research Article
On Hyperbolic 3-Manifolds Obtained by Dehn Surgery on Links
Soo Hwan Kim and Yangkok Kim
Department of Mathematics, Doneeui University, Pusan 614-714, Republic of Korea
Correspondence should be addressed to Yangkok Kim,[email protected] Received 27 July 2010; Accepted 22 September 2010
Academic Editor: S. M. Gusein-Zade
Copyrightq2010 S. H. Kim and Y. Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the algebraic and geometric structures for closed orientable 3-manifolds obtained by Dehn surgery along the family of hyperbolic links with certain surgery coefficients and moreover, the geometric presentations of the fundamental group of these manifolds. We prove that our surgery manifolds are 2-fold cyclic covering of 3-sphere branched over certain link by applying the Montesinos theorem in Montesinos-Amilibia 1975. In particular, our result includes the topological classification of the closed 3-manifolds obtained by Dehn surgery on the Whitehead link, according to Mednykh and Vesnin1998, and the hyperbolic linkLd1ofd1 components in Cavicchioli and Paoluzzi2000.
1. Introduction
All manifolds will be assumed to be connected, orientable, and PL Piecewise Linear. In 1,2, theorems state that any closed orientable 3-manifold can be obtained by Dehn surgeries on the components of an oriented link in the 3-sphere. Considering the hyperbolic link, the Thurston-Jorgensen theory in3of hyperbolic surgery implies that the resulting manifolds are hyperbolic for almost all surgery coefficients. Another method for describing closed 3- manifolds says that any closed 3-manifold can be represented as a branched covering of some link in the 3-sphere2. As the above, if the link is hyperbolic, the construction yields hyperbolic manifolds for branching indices sufficiently large.
According to the algorithm in 4, any manifold obtained by Dehn surgeries on a strongly invertible link can be presented as a 2-fold covering of the 3-sphere branched over some link. Thus we can construct many classes of closed orientable 3-manifolds by considering its branched coverings or by performing Dehn surgery along it. Moreover, the branched covering and Dehn surgery are nice methods for representing closed orientable
L1
L2
L3
Ld
···
Figure 1: The oriented linksLm,dinS3.
3-manifolds by combinatorial tools. See5for the many faces of cyclic branched coverings of 2-bridge links.
In this paper, we consider a family of linksLm,dfor positive integersmanddas in Figure 1, where eachLi in box denotes the 1/m-rational tangle. In fact the linkLm,d has two component links ifdandmare odd, and three components ifdandmis even and odd, respectively. Moreover,Lm,dhasd1-component links ifmis even. ActuallyL1,1is the double link andL2,1 is the Whitehead link which was considered in6, andL2,d is the hyperbolic linkLd1as considered in7. We note thatLm,1is the hyperbolic link form >1 8and thatLm,dis the hyperbolic link form >1, which act by isometries.
Lastly, for positive integersm >1,n≥3, andk≥1, it was proved that a family of closed 3-manifoldM2m1, n, kas the identification space of certain polyhedronP2m1, n, k whose finitely many boundary faces are glued together in pair and which is another method to construct 3-manifolds, is then/d-fold strongly cyclic covering of the 3-sphere branched over the linkLm,d, where gcdn, k d8. Since our linkLm,dis hyperbolic link form >1, it is clear thatM2m1, n, kis the closed hyperbolic 3-manifold.
In this paper, we study the closed hyperbolic 3-manifolds obtained by Dehn surgeries on the components of these links. Moreover, we show that our surgery manifolds are 2-fold cyclic covering of 3-sphere branched over certain link as Figure 6. In particular, our result includes the topological classification of the closed 3-manifolds obtained by Dehn surgery on the Whitehead link, due to Mednykh and Vesnin9, and a hyperbolic linkLd1 ofd1 components in7, which extends the Whitehead link in case ofd 1. See5for similar results obtained by Dehn surgery on the 2-bridge links.
2. Dehn Surgery on the Link L
m,dWe now consider the oriented link Lm,d in the 3-sphere illustrated in Figure 2, which is formed by a chain of linesKibetweenLiandLi1fori1, . . . , d−1, and a chain ofKd, plus a further circleΛtransversally linked toKd. Letpi/ribe the surgery coefficient along theith Ki of the chain fori 1, . . . , d, and let a/bbe the surgery coefficient along the transversal componentΛ, where gcdpi, ri gcda, b 1.
L3
···
yd−1 xd−1
Kd−1
Ld
y′d
x′d u v yd
xd
Kd
L1
x1 y1
K1
L2
x2
y2
K2
zi,m−2
zi,2
zi,1
Li
.. .
Λ
Figure 2:Lm,dwith surgery coefficients andLiwith1/m-rational tangle
On the other hand, we obtain the fundamental groupπ1S3\ Lm,dformeven. Let xi,yi, zi,j,xd,yd,u,and vbe the generators of a Wirtinger presentation ofπ1S3\ Lm,d according toFigure 2. Then we have
π1
S3\ Lm,d
G|R1, R2, R3, 2.1
whereG{x1, . . . , xd, xd, y1, . . . , yd, yd, u, v, zi,j |1≤i≤d,1≤j ≤m−2}andR1, R2, andR3
are as follows under modn;
R1
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩
xi−1x−1i x−1i−1z−1i,11, zi,1xi−1z−1i,1z−1i,21, zi,2zi−1z−1i,2z−1i,3 1,
...
zi,m−2zi,m−3z−1i,m−2y−1i 1, yizi,m−2y−1i yi−11,
fori1, . . . , d−1,
R2
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩ xd−1
xd −1xd−1−1 z−1d,11, zd,1xd−1z−1d,1z−1d,21, zd,2zd−1z−1d,2z−1d,31,
...
zd,m−2zd,m−3z−1d,m−2
yd −11, ydzd,m−3
yd −1yd−11,
R3
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎩
xd−1u−1xdu1, yd−1u−1ydu1, uydv−1
yd −11, xd −1vxdu−11.
2.2
For our simplicity, we write
Cm1i Cm1
x−1i , xi−1
, DmCm
xd, xd−1 , 2.3
where, fori≥3,
C1a, b abbab−1, C2a, b babbaba−1b−1,
Cia, b Ci−2a, bCi−1a,b
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
βaβ−1 andβbaba · · ·ab
i-factors
ifiis odd,
γbγ−1 andγbaba · · ·ba
i-factors
ifiis even.
2.4
Then sincex−1i xi−1 zi,1,zi,2 xi−1zi,1,zi,3 zzi−1i,2inR1, R1reduces to
R1
⎧⎨
⎩
yiCm1,
y−1i−1yiCmy−1i , 1≤i≤d−1. 2.5
SimilarlyR2reduces to
R2
⎧⎨
⎩
ydDm1, y−1d−1ydDm
yd −1. 2.6
Hence we have
π1
S3\ Lm,d
G|R1, R2, R3
, 2.7
where G {x1, . . . , xd, xd, y1, . . . , yd, yd, u, v} and R1, R2, and R3 are as above. We denote byMp1/r1, . . . , pd/rd;a/bthe closed connected orientable 3-manifold obtained by Dehn surgeries along the components K1, K2, . . . , Kd,Λ of Lm,d with surgery coefficients pi/ri, p2/r2, . . . , pd/rd, anda/b, respectively, wherepi, ri a, b 1.
We now obtain finite presentations of the fundamental group of these surgery manifolds as follows.
The meridiansmiand the longitudeliof each componentKiand the meridianmand the longitudelofΛare as follows:
R4
⎧⎨
⎩
mixi i1, . . . , d−1, mu,
R5
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎩
liCmiCm−2i· · ·C2ixi−1C1i1C3i1· · ·Cm−1i1yi1 i1, . . . , d−2,
ld−1 Cmd−1Cm−2d−1· · ·C2d−1xd−2D1D3· · ·Dm−1yd,
ldu−1DmDm−2· · ·D2xd−1C11C31C51· · ·Cm−11y1,
l
yd −1xd.
2.8 A presentation of the fundamental group ofMp1/r1, . . . , pd/rd;a/bis obtained from that of the link group ofLm,dby adding relations:
R6
⎧⎨
⎩
mpiiliri 1 i1, . . . , d,
malb1. 2.9
Sincepiandriresp.,aandbare coprime, there exist integerssiandqiresp.sandqsuch thatrisi−piqi1,for anyi1, . . . , d,andbs−aq1.
Summarizing we obtain the following result.
Theorem 2.1. The fundamental group of the closed connected orientable 3-manifold Mp1/r1, . . . , pd/rd;a/bobtained by Dehn surgery along the linkLm,d with surgery coefficients pi/rianda/badmits the finite presentationG|R1, . . . , R6whereGandRiare as above.
We note that the linkLm,d is hyperbolic in the sense that it has hyperbolic comple- ment. So the Thurston-Jorgensen theory in3of hyperbolic surgery yields the following.
Corollary 2.2. For any integerd≥1, and for almost all pairs of surgery coefficientspi/rianda/b, the closed connected orientable 3-manifoldsMp1/r1, . . . , pd/rd;a/bis hyperbolic.
We now describe Mp1/r1, . . . , pd/rd;a/b as 2-fold branched coverings of the 3- sphere in the following. We note that a linkLis strongly invertible if there is an orientation- preserving involution ofS3which induces on each component ofLan involution with two fixed points. The above mentioned involution is called a strongly invertible involution of the link. The following theorem of Montesinos relates to two different approaches for describing closed orientable 3-manifolds, which is Dehn surgery and branched coveringssee1,10–13 for manuscripts, and moreover, gives us an effective algorithm for describing the branch set ofMp1/r1, . . . , pd/rd;a/bas 2-fold branched coverings of the 3-sphere.
Theorem 2.3see 4. LetM be a closed orientable 3-manifold obtained by Dehn surgery on a strongly invertible linkLofncomponents. ThenMis a 2-fold covering of the 3-sphere branched over a link of at mostn1 components. Conversely, every 2-fold cyclic branched covering of the 3-sphere can be obtained in this fashion.
1 m
1 m
1 m
1 m b
ρ
··· ······
Figure 3: The strongly invertible linksLm, dand its involution.
λ1
λ2 λ3
λd−1
λd
L1
L2
L3
Ld
ρ ρ
b-times
Ld
μi−1 μd−1
b+1 2
···
· · ·
Li
(i=1, . . . , d−1)
Figure 4: Regular neighborhood ofLm,dand strongly involution.
We now apply Dehn surgery on the link Lm,d K1 ∪· · ·∪Kd ∪Λ with surgery coefficientsp1/r1, . . . , pd/rdanda/bfora1,respectively. Twist the solid torusS3\intN, whereNis a tabular neighborhood of the transversal circleΛ.The merideanmofNis carried toτm being the longitude ofN, whereτ represents the number of twists which is positiveresp. negativeif the twist is in the right-handresp. left-handsense. LetLm,d
K1 ∪· · ·∪Kd∪Λbe the link obtained from Lm,d by twisting aroundΛ.Then the surgery
coefficients on componentsΛandKiofLm,dare 1/τbandpi/riforlkΛ, Ki 0. By settingτ : −b,we can delete the componentΛ fromLm,d obtaining the linkLm,d of d components illustrated inFigure 3. This link is strongly invertible, and the axis of a strongly invertible involutionρofLm,dis given by the dotted line inFigure 3.
We choose merideanμi and longitude λi according to Figure 4. LetV be a regular neighborhood of Ld in S3. Without loss of generality, we can choose neighborhood V, meridean μi, and longitudesλi on ∂V to be invariant under the involution ρ. The image ofV under the canonical projectionπ :S3 to the quotient spaceS3/ρofS3underρconsists of d3-balls Bi.Let θ denote the axis of the involution ρ in S3. For each 3-ball Bi, the set
T1
T1
T1
T1
T2
T3 B1
B2 B3
Bd
Bd−1
······
Figure 5: Tangle decompositions.
······ ···
T1
T1
T1
T1
T2
T3 p2
r2
p1
r1
p3
r3
pd−1
rd−1
pd
rd
Figure 6: The branched links obtained by the Montesinos algorithm.
Bi∩πθconsists of two arcs. By isotopy of Bi along the image πλiof longitude λi for anyi1, . . . , d,we getFigure 5. Each 3-ballBi with arcsBi∩πθis a trivial tangle. By the Montesinos algorithm, we replace these trivial tanglesBi bypi/ri-rational tangles for any i1, . . . , d.
For the simplicity, we now define some series of links. We recall that any link can be obtained as the closure of some braid. Given coprime integerspandq,denote byσ2p/q, the rational linkp/q-tangle whose incoming arc are theith link andi1th strings, whereT1, T2
denotes a 4-strings braidσ2−1σ1−1σ3−1σ2−1m and a 3-strings braidσ1σ2σ1σ2σ1b respectively, andT3is a rationalb1/2-tangle.
Summarizing we have proved the following.
Theorem 2.4. Let Mp1/r1, . . . , pd/rd;a/b, d ≥ 2, a ±1, be the closed orientable 3- manifold obtained by Dehn surgery on the link Lm,d with surgery coefficients pi/ri and a/b.
Then Mp1/r1, . . . , pd/rd;a/b is a 2-fold covering of the 3-sphere S3 branched over the link in Figure 6.
Fora/b1 andd2, our manifolds are homeomorphic to the manifolds obtained by Dehn surgeries on the two components of the Whitehead linkL2,1. Thus the result includes the main result in9. Moreover, form 2 andd >1, our manifolds are homeomorphic to the manifolds obtained by Dehn surgeries on thed1components of the hyperbolic link Ld1. In particular the result also includes the result in Section 5 of7.
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