On Hyperbolic 3-Manifolds Obtained by Dehn Surgery on Linksdownloads.hindawi.com/journals/ijmms/2010/573403.pdf · topological classiﬁcation of the closed 3-manifolds obtained by

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2010, Article ID 573403, 8 pagesdoi:10.1155/2010/573403

Research ArticleOn Hyperbolic 3-Manifolds Obtained byDehn 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

Copyright q 2010 S. H. Kim and Y. Kim. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We study the algebraic and geometric structures for closed orientable 3-manifolds obtained byDehn 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 oursurgery manifolds are 2-fold cyclic covering of 3-sphere branched over certain link by applyingthe Montesinos theorem in Montesinos-Amilibia (1975). In particular, our result includes thetopological classification of the closed 3-manifolds obtained by Dehn surgery on the Whiteheadlink, according to Mednykh and Vesnin (1998), and the hyperbolic link Ld+1 of d + 1 componentsin Cavicchioli and Paoluzzi (2000).

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 surgerieson the components of an oriented link in the 3-sphere. Considering the hyperbolic link, theThurston-Jorgensen theory in [3] of hyperbolic surgery implies that the resulting manifoldsare 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 ofsome link in the 3-sphere [2]. As the above, if the link is hyperbolic, the construction yieldshyperbolic manifolds for branching indices sufficiently large.

According to the algorithm in [4], any manifold obtained by Dehn surgeries on astrongly invertible link can be presented as a 2-fold covering of the 3-sphere branchedover some link. Thus we can construct many classes of closed orientable 3-manifolds byconsidering its branched coverings or by performing Dehn surgery along it. Moreover,the branched covering and Dehn surgery are nice methods for representing closed orientable

2 International Journal of Mathematics and Mathematical Sciences

L1

L2

L3

Ld

· · ·Figure 1: The oriented links L(m,d) in S3.

3-manifolds by combinatorial tools. See [5] for the many faces of cyclic branched coveringsof 2-bridge links.

In this paper, we consider a family of links L(m,d) for positive integers m and d as inFigure 1, where each Li in box denotes the 1/m-rational tangle. In fact the link L(m,d) hastwo component links if d and m are odd, and three components if d and m is even and odd,respectively. Moreover, L(m,d) has (d + 1)-component links if m is even. Actually L(1,1) is thedouble link and L(2,1) is the Whitehead link which was considered in [6], and L(2,d) is thehyperbolic link Ld+1 as considered in [7]. We note that L(m,1) is the hyperbolic link for m > 1[8] and that L(m,d) is the hyperbolic link for m > 1, which act by isometries.

Lastly, for positive integersm > 1, n ≥ 3, and k ≥ 1, it was proved that a family of closed3-manifold M(2m + 1, n, k) as the identification space of certain polyhedron P(2m + 1, n, k)whose finitely many boundary faces are glued together in pair and which is another methodto construct 3-manifolds, is the (n/d)-fold strongly cyclic covering of the 3-sphere branchedover the linkL(m,d), where gcd(n, k) = d [8]. Since our linkL(m,d) is hyperbolic link form > 1,it is clear thatM(2m + 1, n, k) is the closed hyperbolic 3-manifold.

In this paper, we study the closed hyperbolic 3-manifolds obtained by Dehn surgerieson the components of these links. Moreover, we show that our surgery manifolds are 2-foldcyclic covering of 3-sphere branched over certain link as Figure 6. In particular, our resultincludes the topological classification of the closed 3-manifolds obtained by Dehn surgery onthe Whitehead link, due to Mednykh and Vesnin [9], and a hyperbolic link Ld+1 of (d + 1)components in [7], which extends the Whitehead link in case of d = 1. See [5] for similarresults obtained by Dehn surgery on the 2-bridge links.

2. Dehn Surgery on the Link L(m,d)

We now consider the oriented link L(m,d) in the 3-sphere illustrated in Figure 2, which isformed by a chain of lines Ki between Li and Li+1 for i = 1, . . . , d − 1, and a chain of Kd, plusa further circle Λ transversally linked to Kd. Let pi/ri be the surgery coefficient along the ithKi of the chain for i = 1, . . . , d, and let a/b be the surgery coefficient along the transversalcomponent Λ, where gcd(pi, ri) = gcd(a, b) = 1.

International Journal of Mathematics and Mathematical Sciences 3

L3

· · ·

yd−1 xd−1

Kd−1

Ldy′d

x′d

u v

yd

xdKd

L1

x1 y1

K1

L2

x2

y2

K2zi,m−2

zi,2

zi,1

Li

...

Λ

Figure 2: L(m,d) with surgery coefficients and Li with (1/m)-rational tangle

On the other hand, we obtain the fundamental group π1(S3 \ L(m,d)) for m even. Letxi, yi, zi,j , x′

d, y′d, u, and v be the generators of a Wirtinger presentation of π1(S3 \ L(m,d))

according to Figure 2. Then we have

π1

(S3 \ L(m,d)

)� 〈G | R1, R2, R3〉, (2.1)

where G = {x1, . . . , xd, x′d, y1, . . . , yd, y

′d, u, v, zi,j | 1 ≤ i ≤ d, 1 ≤ j ≤ m − 2} and R1, R2, and R3

are as follows under modn;

R1 =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

xi−1x−1i x−1

i−1z−1i,1 = 1,

zi,1xi−1z−1i,1z−1i,2 = 1,

zi,2zi−1z−1i,2z−1i,3 = 1,

...

zi,m−2zi,m−3z−1i,m−2y−1i = 1,

yizi,m−2y−1i yi−1 = 1,

for i = 1, . . . , d − 1,

R2 =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

xd−1(x′d

)−1x−1d−1z

−1d,1 = 1,

zd,1xd−1z−1d,1z−1d,2 = 1,

zd,2zd−1z−1d,2z−1d,3 = 1,

...

zd,m−2zd,m−3z−1d,m−2(y′d

)−1 = 1,

y′dzd,m−3

(y′d

)−1yd−1 = 1,


R3 =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

x−1du−1x′

du = 1,

y−1d u−1y′

du = 1,

uy′dv

−1(y′d

)−1 = 1,(x′d

)−1vx′

du−1 = 1.

(2.2)

For our simplicity, we write

Cm+1(i) = Cm+1

(x−1i , xi−1

), Dm = Cm

(x′d, xd−1

), (2.3)

where, for i ≥ 3,

C1(a, b) = ab = bab−1,

C2(a, b) = (b)(ab) = baba−1b−1,

Ci(a, b) = (Ci−2(a, b))Ci−1(a,b) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

βaβ−1 and β = baba · · ·ab︸︷︷︸i-factors

if i is odd,

γbγ−1 and γ = baba · · · ba︸︷︷︸i-factors

if i is even.

(2.4)

Then since (x−1i )xi−1 = zi,1, zi,2 = x

zi,1i−1, zi,3 = z

zi,2i−1 in R1, R1 reduces to

R′1 =

⎧⎨⎩yi = Cm+1,

y−1i−1 = yiCmy

−1i ,

1 ≤ i ≤ d − 1. (2.5)

Similarly R2 reduces to

R′2 =

⎧⎨⎩y′d= Dm+1,

y−1d−1 = y′

dDm

(y′d

)−1.

(2.6)

Hence we have

π1

(S3 \ L(m,d)

)� ⟨

G | R′1, R

′2, R3

⟩, (2.7)

where G = {x1, . . . , xd, x′d, y1, . . . , yd, y

′d, u, v} and R′

1, R′2, and R3 are as above. We

denote by M(p1/r1, . . . , pd/rd;a/b) the closed connected orientable 3-manifold obtained byDehn surgeries along the components K1, K2, . . . , Kd,Λ of L(m,d) with surgery coefficientspi/ri, p2/r2, . . . , pd/rd, and a/b, respectively, where (pi, ri) = (a, b) = 1.

We now obtain finite presentations of the fundamental group of these surgerymanifolds as follows.


The meridians mi and the longitude li of each component Ki and the meridian m andthe longitude l of Λ are as follows:

R4 =

⎧⎨⎩mi = xi (i = 1, . . . , d − 1),

m = u,

R5 =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

li = Cm(i)Cm−2(i) · · ·C2(i)xi−1C1(i+1)C3(i+1) · · ·Cm−1(i+1)yi+1 (i = 1, . . . , d−2),ld−1 = Cm(d − 1)Cm−2(d − 1) · · ·C2(d − 1)xd−2D1D3 · · ·Dm−1y′

d,

ld = u−1DmDm−2 · · ·D2xd−1C1(1)C3(1)C5(1) · · ·Cm−1(1)y1,

l =(y′d

)−1x′d.

(2.8)

A presentation of the fundamental group ofM(p1/r1, . . . , pd/rd;a/b) is obtained from that ofthe link group of L(m,d) by adding relations:

R6 =

⎧⎨⎩m

pii l

rii = 1 (i = 1, . . . , d),

malb = 1.(2.9)

Since pi and ri (resp., a and b) are coprime, there exist integers si and qi (resp. s and q) suchthat risi − piqi = 1, for any i = 1, . . . , d, and bs − aq = 1.

Summarizing we obtain the following result.

Theorem 2.1. The fundamental group of the closed connected orientable 3-manifoldM(p1/r1, . . . , pd/rd;a/b) obtained by Dehn surgery along the link L(m,d) with surgery coefficientspi/ri and a/b admits the finite presentation 〈G | R1, . . . , R6〉 where G and Ri are as above.

We note that the link L(m,d) is hyperbolic in the sense that it has hyperbolic comple-ment. So the Thurston-Jorgensen theory in [3] of hyperbolic surgery yields the following.

Corollary 2.2. For any integer d ≥ 1, and for almost all pairs of surgery coefficients pi/ri and a/b,the closed connected orientable 3-manifolds M(p1/r1, . . . , pd/rd;a/b) is hyperbolic.

We now describe M(p1/r1, . . . , pd/rd;a/b) as 2-fold branched coverings of the 3-sphere in the following. We note that a link L is strongly invertible if there is an orientation-preserving involution of S

3 which induces on each component of L an involution with twofixed points. The above mentioned involution is called a strongly invertible involution of thelink. The following theorem of Montesinos relates to two different approaches for describingclosed orientable 3-manifolds, which is Dehn surgery and branched coverings (see [1, 10–13]for manuscripts), and moreover, gives us an effective algorithm for describing the branch setofM(p1/r1, . . . , pd/rd;a/b) as 2-fold branched coverings of the 3-sphere.

Theorem 2.3 (see [4]). Let M be a closed orientable 3-manifold obtained by Dehn surgery on astrongly invertible link L of n components. ThenM is a 2-fold covering of the 3-sphere branched overa link of at most n + 1 components. Conversely, every 2-fold cyclic branched covering of the 3-spherecan be obtained in this fashion.


1m

1m

1m

1m

b

ρ

· · ·· · ·· · ·

Figure 3: The strongly invertible links L′′(m,d) and its involution.

λ1

λ2λ3

λd−1

λdL1

L2

L3

Ld

ρ ρ

b-times

Ld

μi−1 μd−1

b + 12

· · ·

· · ·Li

(i = 1, . . . , d − 1)

Figure 4: Regular neighborhood of L′′(m,d) and strongly involution.

We now apply Dehn surgery on the link L(m,d) = K1 ∪ · · · ∪ Kd ∪ Λ with surgerycoefficients p1/r1, . . . , pd/rd and a/b for a = 1, respectively. Twist the solid torus S

3 \ int(N),whereN is a tabular neighborhood of the transversal circleΛ. Themerideanm ofN is carriedto τ� + m (� being the longitude of N), where τ represents the number of twists which ispositive (resp. negative) if the twist is in the right-hand (resp. left-hand) sense. Let L′

(m,d) =K′

1 ∪ · · · ∪ K′d ∪ Λ′ be the link obtained from L(m,d) by twisting around Λ. Then the surgery

coefficients on components Λ′ and K′i of L′

(m,d) are 1/(τ + b) and pi/ri for lk(Λ, Ki) = 0. Bysetting τ := −b, we can delete the component Λ′ from L′

(m,d) obtaining the link L′′(m,d) of d

components illustrated in Figure 3. This link is strongly invertible, and the axis of a stronglyinvertible involution ρ of L′′

(m,d) is given by the dotted line in Figure 3.We choose meridean μi and longitude λi according to Figure 4. Let V be a regular

neighborhood of L′′din S

3. Without loss of generality, we can choose neighborhood V,meridean μi, and longitudes λi on ∂V to be invariant under the involution ρ. The imageof V under the canonical projection π : S3 to the quotient space S3/ρ of S3 under ρ consistsof d 3-balls Bi. Let θ denote the axis of the involution ρ in S3. For each 3-ball Bi, the set


T1

T1

T1T1

T2

T3

B1

B2 B3

Bd

Bd−1

· · ·· · ·

· · ·

Figure 5: Tangle decompositions.

· · ·· · ·· · ·

T1

T1

T1T1

T2

T3

p2r2

p1r1

p3r3

pd−1rd−1

pdrd

Figure 6: The branched links obtained by the Montesinos algorithm.

Bi ∩ π(θ) consists of two arcs. By isotopy of Bi along the image π(λi) of longitude λi forany i = 1, . . . , d, we get Figure 5. Each 3-ball Bi with arcs Bi ∩ π(θ) is a trivial tangle. By theMontesinos algorithm, we replace these trivial tangles Bi by (pi/ri)-rational tangles for anyi = 1, . . . , d.

For the simplicity, we now define some series of links. We recall that any link can beobtained as the closure of some braid. Given coprime integers p and q, denote by σ

p/q

2 , therational link (p/q)-tangle whose incoming arc are the ith link and (i+1)th strings, where T1, T2denotes a 4-strings braid (σ−1

2 σ−11 σ−1

3 σ−12 )m and a 3-strings braid (σ1σ2σ1σ2σ1)

b respectively,and T3 is a rational ((b + 1)/2)-tangle.

Summarizing we have proved the following.

Theorem 2.4. Let M(p1/r1, . . . , pd/rd;a/b), d ≥ 2, a = ±1, be the closed orientable 3-manifold obtained by Dehn surgery on the link L(m,d) with surgery coefficients pi/ri and a/b.Then M(p1/r1, . . . , pd/rd;a/b) is a 2-fold covering of the 3-sphere S

3 branched over the link inFigure 6.


For a/b = 1 and d = 2, our manifolds are homeomorphic to the manifolds obtained byDehn surgeries on the two components of the Whitehead link L(2,1). Thus the result includesthe main result in [9]. Moreover, for m = 2 and d > 1, our manifolds are homeomorphic tothe manifolds obtained by Dehn surgeries on the (d + 1) components of the hyperbolic linkLd+1. In particular the result also includes the result in Section 5 of [7].

References

[1] W. B. R. Lickorish, “A representation of orientable combinatorial 3-manifolds,” Annals of Mathematics,vol. 76, pp. 531–540, 1962.

[2] D. Rolfsen, Knots and Links, Math.Lect. Series 7, Publish or Perish, Berkeley, Calif, USA, 1978.[3] W. P. Thurston, The Geometry and Topology of 3-Manifolds, Lect. Notes, Princeton University Press,

Princeton, NJ, USA, 1980.[4] J. M. Montesinos, “Surgery on links and double branched covers of S3,” in knots, Groups and 3-

Manifolds, L. P. Neuwirth, Ed., pp. 227–259, Princeton University Press, Princeton, NJ, USA, 1975.[5] M. Mulazzani and A. Vesnin, “The many faces of cyclic branched coverings of 2-bridge knots and

links,” Atti del Seminario Matematico e Fisico dell’Universita di Modena, vol. 49, pp. 177–215, 2001.[6] H. Helling, A. C. Kim, and J. L. Mennicke, “Some honey-combs in hyperbolic 3-space,” Communica-

tions in Algebra, vol. 23, no. 14, pp. 5169–5206, 1995.[7] A. Cavicchioli and L. Paoluzzi, “On certain classes of hyperbolic 3-manifolds,” Manuscripta

Mathematica, vol. 101, no. 4, pp. 457–494, 2000.[8] S. H. Kim and Y. Kim, “On certain classes of links and 3-manifolds,” Communications of the Korean

Mathematical Society, vol. 20, no. 4, pp. 803–812, 2005.[9] A. Mednykh and A. Vesnin, “Covering properties of small volume hyperbolic 3-manifolds,” Journal

of Knot Theory and Its Ramifications, vol. 7, no. 3, pp. 381–392, 1998.[10] J. S. Birman and H. M. Hilden, “Heegaard splittings of branched coverings of S3,” Transactions of the

American Mathematical Society, vol. 213, pp. 315–352, 1975.[11] R. C. Kirby, “A calculus for framed links in S3,” Inventiones Mathematicae, vol. 45, pp. 315–352, 1978.[12] J. M. Montesinos and W. Whitten, “Constructions of two-fold branched covering spaces,” Pacific

Journal of Mathematics, vol. 125, no. 2, pp. 415–446, 1986.[13] B. Zimmermann, “On cyclic branched coverings of hyperbolic links,” Topology and Its Applications,

vol. 65, no. 3, pp. 287–294, 1995.

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On Hyperbolic 3-Manifolds Obtained by Dehn Surgery on Linksdownloads.hindawi.com/journals/ijmms/2010/573403.pdf · topological classiﬁcation of the closed 3-manifolds obtained by