On Hyperbolic 3-Manifolds Obtained by Dehn Surgery on Links

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 Vesnin 1998 , and the hyperbolic link Ld 1 of d 1 components in Cavicchioli and Paoluzzi 2000 .


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 3manifolds 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 International Journal of Mathematics and Mathematical Sciences 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 links L m,d for positive integers m and d as in Figure 1, where each L i 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 L d 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 integers m > 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 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 Vesnin 9 , and a hyperbolic link L d 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.

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 is formed by a chain of lines K i between L i and L i 1 for i 1, . . ., d − 1, and a chain of K d , plus a further circle Λ transversally linked to K d .Let p i /r i be the surgery coefficient along the ith K i of the chain for i 1, . . ., d, and let a/b be the surgery coefficient along the transversal component Λ, where gcd p i , r i gcd a, b 1.On the other hand, we obtain the fundamental group π 1 S 3 \ L m,d for m even.Let x i , y i , z i,j , x d , y d , u, and v be the generators of a Wirtinger presentation of π 1 S 3 \ L m,d according to Figure 2. Then we have , and R 3 are as follows under modn; 1,

2.2
For our simplicity, we write where, for i ≥ 3, Similarly R 2 reduces to Hence we have where G {x 1 , . . ., x d , x d , y 1 , . . ., y d , y d , u, v} and R 1 , R 2 , and R 3 are as above.We denote by M p 1 /r 1 , . . ., p d /r d ; a/b the closed connected orientable 3-manifold obtained by Dehn surgeries along the components K 1 , K 2 , . . ., K d , Λ of L m,d with surgery coefficients p i /r i , p 2 /r 2 , . . ., p d /r d , and a/b, respectively, where p i , r i a, b 1.We now obtain finite presentations of the fundamental group of these surgery manifolds as follows.
The meridians m i and the longitude l i of each component K i and the meridian m and the longitude l of Λ are as follows:

2.8
A presentation of the fundamental group of M p 1 /r 1 , . . ., p d /r d ; a/b is obtained from that of the link group of L m,d by adding relations:

2.9
Since p i and r i resp., a and b are coprime, there exist integers s i and q i resp.s and q such that r i s i − p i q i 1, for any i 1, . . ., d, and bs − aq 1.
Summarizing we obtain the following result.We note that the link L 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 integer d ≥ 1, and for almost all pairs of surgery coefficients p i /r i and a/b, the closed connected orientable 3-manifolds M p 1 /r 1 , . . ., p d /r d ; a/b is hyperbolic.
We now describe M p 1 /r 1 , . . ., p d /r d ; a/b as 2-fold branched coverings of the 3sphere in the following.We note that a link L is strongly invertible if there is an orientationpreserving involution of S 3 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 p 1 /r 1 , . . ., p d /r d ; 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 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.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.
We choose meridean μ i and longitude λ i according to Figure 4. Let V be a regular neighborhood of L d in S 3 .Without loss of generality, we can choose neighborhood V, meridean μ i , and longitudes λ i on ∂V to be invariant under the involution ρ.The image of V under the canonical projection π : S 3 to the quotient space S 3 /ρ of S 3 under ρ consists of d 3-balls B i .Let θ denote the axis of the involution ρ in S 3 .For each 3-ball B i , the set B i ∩ π θ consists of two arcs.By isotopy of B i along the image π λ i of longitude λ i for any i 1, . . ., d, we get Figure 5.Each 3-ball B i with arcs B i ∩ π θ is a trivial tangle.By the Montesinos algorithm, we replace these trivial tangles B i by p i /r i -rational tangles for any i 1, . . ., 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 integers p and q, denote by σ p/q 2 , the rational link p/q -tangle whose incoming arc are the ith link and i 1 th strings, where T 1 , T 2 denotes a 4-strings braid σ −1 2 σ −1

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

Theorem 2 . 1 .
The fundamental group of the closed connected orientable 3-manifold M p 1 /r 1 , . . ., p d /r d ; a/b obtained by Dehn surgery along the link L m,d with surgery coefficients p i /r i and a/b admits the finite presentation G | R 1 , . . ., R 6 where G and R i are as above.

Figure 3 :
Figure 3: The strongly invertible links L m, d and its involution. λ

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

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

2 m and a 3 -Theorem 2 . 4 .
strings braid σ 1 σ 2 σ 1 σ 2 σ 1 b respectively, and T 3 is a rational b 1 /2 -tangle.Summarizing we have proved the following.Let M p 1 /r 1 , . . ., p d /r d ; a/b , d ≥ 2, a ±1, be the closed orientable 3manifold obtained by Dehn surgery on the link L m,d with surgery coefficients p i /r i and a/b.Then M p 1 /r 1 , . . ., p d /r d ; a/b is a 2-fold covering of the 3-sphere S 3 branched over the link in Figure 6.