# On Hyperbolic 3-Manifolds Obtained by Dehn Surgery on Links

Authors: Soo Hwan Kim; Yangkok Kim
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

We study the algebraic and geometric structures for
closed orientable3-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 Vesnin (1998), and
the hyperbolic link Ld+1 of d+1 components in Cavicchioli and Paoluzzi (2000).


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## Body

## 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 in [3] of 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-sphere [2]. 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 3-manifolds by combinatorial tools. See [5] for the many faces of cyclic branched coverings of 2-bridge links.In this paper, we consider a family of linksℒ(m,d) for positive integers m and d as in Figure 1, where each Li in box denotes the 1/m-rational tangle. In fact the link ℒ(m,d) has two 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 the double link and ℒ(2,1) is the Whitehead link which was considered in [6], and ℒ(2,d) is the hyperbolic 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.Figure 1
The oriented linksℒ(m,d) in S3.Lastly, for positive integersm>1, n≥3, and k≥1, it was proved that a family of closed 3-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 method to construct 3-manifolds, is the (n/d)-fold strongly cyclic covering of the 3-sphere branched over the link ℒ(m,d), where gcd(n,k)=d [8]. Since our link ℒ(m,d) is hyperbolic link for m>1, it is clear that ℳ(2m+1,n,k) is the closed hyperbolic 3-manifold.In this paper, we study the closed hyperbolic3-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 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 similar results obtained by Dehn surgery on the 2-bridge links.


## 2. Dehn Surgery on the Linkℒ(m,d)

We now consider the oriented linkℒ(m,d) in the 3-sphere illustrated in Figure 2, which is formed by a chain of lines Ki between Li and Li+1 for i=1,…,d-1, and a chain of Kd, plus a further circle Λ transversally linked to Kd. Let  pi/ri  be the surgery coefficient along the ith Ki of the chain for i=1,…,d, and let  a/b be the surgery coefficient along the transversal component Λ, where gcd(pi,ri)=gcd(a,b)=1.Figure 2
ℒ(m,d) with surgery coefficients and Li with (1/m)-rational tangleOn the other hand, we obtain the fundamental groupπ1(𝕊3∖ℒ(m,d)) for m even. Let xi, yi, zi,j, xd′, yd′, u, and v be the generators of a Wirtinger presentation of π1(𝕊3∖ℒ(m,d)) according to Figure 2. Then we have(2.1)π1(𝕊3∖ℒ(m,d))≃〈G∣R1,R2,R3〉,
where G={x1,…,xd,xd′,y1,…,yd,yd′,u,v,zi,j∣1≤i≤d,1≤j≤m-2} and R1,R2, and R3 are as follows under modn;(2.2)R1={xi-1xi-1xi-1-1zi,1-1=1,zi,1xi-1zi,1-1zi,2-1=1,zi,2zi-1zi,2-1zi,3-1=1,⋮zi,m-2zi,m-3zi,m-2-1yi-1=1,yizi,m-2yi-1yi-1=1,fori=1,…,d-1,R2={xd-1(xd′)-1xd-1-1zd,1-1=1,zd,1xd-1zd,1-1zd,2-1=1,zd,2zd-1zd,2-1zd,3-1=1,⋮zd,m-2zd,m-3zd,m-2-1(yd′)-1=1,yd′zd,m-3(yd′)-1yd-1=1,R3={xd-1u-1xd′u=1,yd-1u-1yd′u=1,uyd′v-1(yd′)-1=1,(xd′)-1vxd′u-1=1.
For our simplicity, we write(2.3)Cm+1(i)=Cm+1(xi-1,xi-1),Dm=Cm(xd′,xd-1),
where, for i≥3,(2.4)C1(a,b)=ab=ba̲b-1,C2(a,b)=(b)(ab)=bab̲a-1b-1,Ci(a,b)=(Ci-2(a,b))Ci-1(a,b)={βaβ-1andβ=baba⋯ab︸i-factorsifiisodd,γbγ-1andγ=baba⋯ba︸i-factorsifiiseven.
Then since (xi-1)xi-1=zi,1, zi,2=xi-1zi,1, zi,3=zi-1zi,2 in R1,R1 reduces to(2.5)R1′={yi=Cm+1,yi-1-1=yiCmyi-1,1≤i≤d-1.
Similarly R2 reduces to (2.6)R2′={yd′=Dm+1,yd-1-1=yd′Dm(yd′)-1.
Hence we have(2.7)π1(𝕊3∖ℒ(m,d))≃〈G∣R1′,R2′,R3〉,
where G={x1,…,xd,xd′,y1,…,yd,yd′,u,v} and R1′,R2′, and R3 are as above.  We denote by ℳ(p1/r1,…,pd/rd;a/b) the closed connected orientable 3-manifold obtained by Dehn surgeries along the components K1,K2,…,Kd,Λ of ℒ(m,d) with surgery coefficients pi/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 surgery manifolds as follows.The meridiansmi and the longitude li of each component Ki and the meridian m and the longitude l of Λ are as follows:
(2.8)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-1yd′,ld=u-1DmDm-2⋯D2xd-1C1(1)C3(1)C5(1)⋯Cm-1(1)y1,l=(yd′)-1xd′.
A presentation of the fundamental group of ℳ(p1/r1,…,pd/rd;a/b) is obtained from that of the link group of ℒ(m,d) by adding relations:(2.9)R6={mipiliri=1(i=1,…,d),malb=1.
Since pi and ri (resp., a and b) are coprime, there exist integers si and qi (resp. s and q) such that 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 coefficients pi/ri and a/b admits the finite presentation 〈G∣R1,…,R6〉  where G and Ri are as above.We note that the linkℒ(m,d) is hyperbolic in the sense that it has hyperbolic complement. So the Thurston-Jorgensen theory in [3] of hyperbolic surgery yields the following.Corollary 2.2.
For any integerd≥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 describeℳ(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 𝕊3 which induces on each component of ℒ an 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 coverings (see [1, 10–13] for manuscripts), and moreover, gives us an effective algorithm for describing the branch set of ℳ(p1/r1,…,pd/rd;a/b) as 2-fold branched coverings of the 3-sphere.Theorem 2.3 (see [4]).
LetM be a closed orientable 3-manifold obtained by Dehn surgery on a strongly invertible link L of n components. Then M is a 2-fold covering of the 3-sphere branched over a link of at most n+1 components. Conversely, every 2-fold cyclic branched covering of the 3-sphere can be obtained in this fashion.We now apply Dehn surgery on the linkℒ(m,d)=K1∪⋯∪Kd∪Λ with surgery coefficients p1/r1,…,pd/rd and a/b for a=1, respectively. Twist the solid torus 𝕊3∖int(N), where N is a tabular neighborhood of the transversal circle Λ. The meridean m of N is carried to τℓ+m (ℓ being the longitude of N), where τ represents the number of twists which is positive (resp. negative) if the twist is in the right-hand (resp. left-hand) sense. Let ℒ(m,d)′=K1′∪⋯∪Kd′∪Λ′ be the link obtained from ℒ(m,d) by twisting around Λ. Then the surgery coefficients on components Λ′ and Ki′ of ℒ(m,d)′ are 1/(τ+b) and pi/ri for lk(Λ,Ki)=0. By setting 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 strongly invertible involution ρ of ℒ(m,d)′′ is given by the dotted line in Figure 3.Figure 3
The strongly invertible linksℒ′′(m,d) and its involution.We choose merideanμi and longitude λi according to Figure 4. Let V be a regular neighborhood of Ld′′ in 𝕊3. Without loss of generality, we can choose neighborhood V, meridean μi, and longitudes λi on ∂V to be invariant under the involution ρ. The image of V under the canonical projection π:S3 to the quotient space S3/ρ of S3 under ρ consists of d3-balls Bi. Let θ denote the axis of the involution ρ in S3. For each 3-ball Bi, the set Bi∩π(θ) consists of two arcs. By isotopy of Bi along the image π(λi) of longitude λi for any i=1,…,d, we get Figure 5. Each 3-ball Bi with arcs Bi∩π(θ) is a trivial tangle. By the Montesinos algorithm, we replace these trivial tangles Bi by (pi/ri)-rational tangles for any i=1,…,d.Figure 4
Regular neighborhood ofℒ(m,d)′′ and strongly involution.Figure 5
Tangle decompositions.Figure 6
The branched links obtained by the Montesinos algorithm.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 integersp and q, denote by σ2p/q, the rational link (p/q)-tangle whose incoming arc are the ith link and (i+1)th strings, where T1,T2 denotes a 4-strings braid (σ2-1σ1-1σ3-1σ2-1)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.
Letℳ(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 𝕊3 branched over the link in Figure 6.Fora/b=1 and d=2, our manifolds are homeomorphic to the manifolds obtained by Dehn surgeries on the two components of the Whitehead link ℒ(2,1). Thus the result includes the main result in [9]. Moreover, for m=2 and d>1, our manifolds are homeomorphic to the manifolds obtained by Dehn surgeries on the (d+1) components of the hyperbolic link Ld+1. In particular the result also includes the result in Section 5 of [7].



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