# Fundamental Group and Covering Properties of Hyperbolic Surgery Manifolds

Authors: Alberto Cavicchioli; Fulvia Spaggiari; Agnese Ilaria Telloni
Journal: Geometry
 (2013)

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## Abstract

We study a family of closed connected orientable 3-manifolds obtained by Dehn
surgeries with rational coefficients along the oriented components of certain links. This
family contains all the manifolds obtained by surgery along the (hyperbolic) 2-bridge
knots. We find geometric presentations for the fundamental group of such manifolds and
represent them as branched covering spaces. As a consequence, we prove that the surgery
manifolds, arising from the hyperbolic 2-bridge knots, have Heegaard genus 2 and are
2-fold coverings of the 3-sphere branched over well-specified links.


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## Body

## 1. Manifolds Obtained by Dehn Surgeries

As well known, any closed connected orientable 3-manifold can be obtained by Dehn surgeries on the components of an oriented link in the 3-sphere (see [1, 2]). If such a link is hyperbolic, then the Thurston-Jorgensen theory [3] of hyperbolic surgery implies that the resulting manifolds are hyperbolic for almost all surgery coefficients. Another method for studying a closed orientable 3-manifold is to represent it as a branched covering of a link in the 3-sphere (see, e.g., [4]). If such a link is hyperbolic, then the construction yields hyperbolic manifolds for branching indices sufficiently large. In the context of current research in 3-manifold topology, many classes of closed orientable hyperbolic 3-manifolds have been constructed by considering branched coverings of links or by performing Dehn surgery along them (see, e.g., [5–10]). This paper relates these methods to study a new class of hyperbolic orientable 3-manifolds via combinatorial tools. More precisely, for any positive integer n, let ℒ2n+1 be the oriented link with 2n+1 components L0, Li, and Ki, i=1,…,n, in the oriented 3-sphere 𝕊3 depicted in Figure 1. This link can be obtained as a belted sum of Borromean rings, as remarked in [11, p. 8]; thus, it is hyperbolic for any n≥1. Let us consider the closed connected orientable 3-manifolds Mn(ri/si;pi/qi;h/k) obtained by Dehn surgery on 𝕊3 along the oriented link ℒ2n+1 such that the surgery coefficients ri/si, pi/qi, and h/k correspond to the oriented components Li, Ki, and L0, respectively, where i=1,…,n. Of course, we always assume that gcd(ri,si)=1, gcd(pi,qi)=1, and gcd(h,k)=1. Here we will show that our family of manifolds contains all closed manifolds obtained by Dehn surgeries on 2-bridge knots. Such manifolds and their geometries were studied in a nice paper of Brittenham and Wu, where the exceptional Dehn surgeries on 2-bridge knots were completely classified (see [5]). This fact gives a further motivation for the study of our surgery manifolds. Recall that a nontrivial Dehn surgery on a hyperbolic knot in the oriented 3-sphere is said to be exceptional if the resulting manifold is either reducible, toroidal, or a Seifert fibered manifold whose orbifold base is the 2-sphere with at most three exceptional fibers (called a small Seifert fibered space). Thus an exceptional Dehn surgery is not hyperbolic. Moreover, it can be shown that a nonexceptional surgery on a 2-bridge knot is hyperbolic (see [5]). Now we determine a geometric presentation for the fundamental group of the surgery manifold Mn(ri/si;pi/qi;h/k). A group presentation is said to be geometric if it arises from a Heegaard diagram of a closed connected (orientable) 3-manifold. If so, then the presentation also corresponds to a spine of the considered manifold. A Wirtinger presentation of the link group π(ℒ2n+1)=π1(𝕊3∖ℒ2n+1) has generators x0, xi, and yi, for every i=1,…,n (see Figure 1).Figure 1
Dehn surgery description of the 3-manifoldMn(ri/si;pi/qi;h/k) and the generators of a Wirtinger presentation of π(ℒ2n+1).The meridiansmi and μi and the longitudes li and λi of the components Li and Ki, respectively, of ℒ2n+1 are
(1)mi=xi,li=yi-1-1xi-1-1⋯y1-1x1-1y1⋯ynx0yn-1⋯y1-1x1y1⋯xi-1yi-1xi-1-1⋯x1-1x0-1x1⋯xi-1,μi=yi,λi=yi⋯ynx0-1yn-1⋯yi-1xi-1yi-1-1⋯x2-1y1-1x1-1y1⋯ynx0yn-1⋯y1-1x1y1x2⋯yi-1xi,
where [mi,li]=1 and [μi,λi]=1 for every i=1,…,n. The meridian m0 and the longitude l0 of the component L0 of ℒ2n+1 are
(2)m0=x0,l0=yn-1⋯y1-1x1y1⋯xnynxn-1⋯x1-1.To determine the formulae for longitudesli, λi, and l0, we have used the following procedure. Fix an orientation and an initial point for each component of the link ℒ2n+1. Starting from the initial point, we run along the component in the sense of the fixed orientation and write in order only the generators encountered at the undercrossings. At each undercrossing we write the generator (represented by the oriented arc running over the undercrossing) with positive (resp., negative) exponent if the sense of percorrence is equal (resp., opposite) to the orientation of the named arc. The obtained longitude is homologous to zero in the complement of the considered component if the exponent sum is equal to zero.A finite presentation for the fundamental group of the surgery manifoldMn(ri/si;pi/qi;h/k) is obtained from that of π(ℒ2n+1) by adding the relations
(3)mirilisi=1,μipiλiqi=1,m0hl0k=1,
for i=1,…,n. Since the integers of the pairs (pi,qi), (ri,si), and (h,k) are coprime, there are integers αi, βi, γi, δi, ξ, and η such that
(4)qiαi-piβi=1,siγi-riδi=1,kξ-hη=1.
Let us define
(5)ai:=miγiliδi,bi:=μiαiλiβi,c:=m0ξl0η,
for i=1,…,n.Then we have(6)aisi=(miγiliδi)si=mimiriδiliδisi=mi(mirilisi)δi=mi=xi,biqi=(μiαiλiβi)qi=μiμipiβiλiqiβi=μi(μipiλiqi)βi=μi=yi,ck=(m0ξl0η)k=m0m0hηl0kη=m0(m0hl0k)η=m0=x0,ai-ri=(miγiliδi)-ri=mi-riγili-siγili=(mirilisi)-γili=li,bi-pi=(μiαiλiβi)-pi=μi-piαiλi-qiαiλi=(μipiλiqi)-αiλi=λi,c-h=(m0ξl0η)-h=m0-hξl0-kξl0=(m0hl0k)-ξl0=l0
for i=1,…,n. We have the following result.Theorem 1.
The fundamental group of the surgery 3-dimensional manifoldMn(ri/si;pi/qi;h/k) admits the finite balanced presentation with 2n+1 generators ai, bi, and c, i=1,…,n, and 2n+1 relations:
(7)airibiqi⋯bnqnckbn-qn⋯bi-qic-k=1,bipic-kai-si⋯a1-s1cka1s1⋯aisi=1,chbn-qn⋯b1-q1a1s1b1q1⋯ansnbnqnan-sn⋯a1-s1=1.
The closed manifoldMn(ri/si;pi/qi;h/k) admits a Heegaard diagram of genus 2n+1 inducing the above presentation, which is thus geometric. Furthermore, the Heegaard genus of Mn(ri/si;pi/qi;h/k) is at most 2n+1.Proof.
Substituting the above relations in the relators of the Wirtinger presentation ofπ(ℒ2n+1) and using the previous formulae for the longitudes li, λi, and l0, we get the relations of the statement. More precisely, substituting l1=a1-r1, yi=biqi, and x0=ck into
(8)l1=y1⋯ynx0yn-1⋯y1-1x0-1
we get
(9)a1-r1=b1q1⋯bnqnckbn-qn⋯b1-q1c-k
or, equivalently,
(10)a1r1b1q1⋯bnqnckbn-qn⋯b1-q1c-k=1
which is the first relation of the statement for i=1. Then we have
(11)b1-p1=λ1=(y1⋯ynx0-1yn-1⋯y1-1)x1-1×(y1⋯ynx0yn-1⋯y1-1)x1=(l1x0)-1x1-1(l1x0)x1=c-kar1a1-s1a1-r1cka1s1=c-ka1-s1cka1s1
or, equivalently,
(12)b1p1c-ka1-s1cka1s1=1
which is the second relation of the statement for i=1. From the expression of l2 we get
(13)a2-r2=l2=y1-1x1-1(y1⋯ynx0yn-1⋯y1-1)x1y1x1-1x0-1x1=b1-q1a1-s1(l1x0)a1s1b1q1a1-s1c-ka1s1=b1-q1a1-s1a1-r1cka1s1b1q1a1-s1c-ka1s1=b1-q1a1-r1(a1-s1cka1s1)b1q1(a1-s1c-ka1s1)=b1-q1a1-r1ckb1-p1b1q1b1p1c-k=b1-q1a1-r1ckb1q1c-k=b1-q1(b1q1⋯bnqnckbn-qn⋯b1-q1c-k)ckb1q1c-k=b2q2⋯bnqnckbn-qn⋯b2-q2c-k
or, equivalently,
(14)a2r2b2q2⋯bnqnckbn-qn⋯b2-q2c-k=1
which is the first relation of the statement for i=2. From the expression of λ2 we get
(15)b2-p2=λ2=(y2⋯ynx0-1yn-1⋯y2-1)x2-1y1-1x1-1×(y1⋯ynx0yn-1⋯y1-1)x1y1x2=(b2q2⋯bnqnc-kbn-qn⋯b2-q2)a2-s2b1-q1a1-s1×(b1q1⋯bnqnckbn-qn⋯b1-q1)a1s1b1q1a2s2=c-ka2r2a2-s2(b1-q1a1-s1a1-r1cka1s1b1q1)a2s2=c-ka2-s2a2r2a2-r2a1-s1cka1s1a2s2=c-ka2-s2a1-s1cka1s1a2s2
or, equivalently,
(16)b2p2c-ka2-s2a1-s1cka1s1a2s2=1
which is the second relation of the statement for i=2. Going on like this, we get by finite iteration the first and second relations of the statement for i=1,…,n. Substituting l0=c-h,  yi=biqi, and xi=aisi into
(17)l0=yn-1⋯y1-1x1y1⋯xnynxn-1⋯x1-1
we get
(18)c-h=bn-qn⋯b1-q1a1s1b1q1⋯ansnbnqnan-sn⋯a1-s1
which gives the last relation of the statement. To show that the presentation in Theorem 1 is geometric, it suffices to draw a suitable RR-system (Rail-Road system) which induces precisely the above presentation (see Figure 2). The hexagons represent the generators, and the three curves labelled by 1, 2, or 3 arrows correspond to the relations in the statement of Theorem 1. For the theory of RR-systems we refer the reader to [12, 13].Figure 2
An RR-system of genus2n+1 inducing the presentation of π1(Mn(ri/si;pi/qi;h/k)).We also note that the first integral homology group ofMn(ri/si;pi/qi;h/k) is isomorphic to ⨁i=1n(ℤ|ri|⊕ℤ|pi|)⊕ℤ|h|. For example, if ri=pi=h=0, i=1,…,n, then the Heegaard genus of our surgery manifolds is exactly 2n+1.As remarked in [11, p. 8], the link ℒ2n+1 is hyperbolic in the sense that it has a hyperbolic complement. So the Thurston-Jorgensen theory [3] of hyperbolic surgery gives the following result.Theorem 2.
For any integern≥1 and for almost all pairs of surgery coefficients ri/si, pi/qi, and h/k, the closed connected orientable 3-manifolds Mn(ri/si;pi/qi;h/k) are hyperbolic.Ifri=pi=1 for every i=1,…,n, then the surgery 3-dimensional manifold Mn(1/si;1/qi;h/k) is homeomorphic to the closed orientable 3-manifold Kα/β(h/k) obtained by h/k Dehn surgery on the 2-bridge knot Kα/β corresponding to the Conway parameters [-2s1,2q1,…,-2sn,2qn], as shown in Figure 3. Note that our parameterization is coherent with that used by Rolfsen [4, p. 303], by setting c1=-2s1, c2=2q1, and so on. The ci in Rolfsen notation indicate the number of crossings and are negative if the sense of the crossings is reversed. This implies that our picture in Figure 3 is slightly different to that drawn in Rolfsen [4, p. 303], as ci and 2si have opposite signs for i odd. In particular, ci is negative for i odd since si≥1. We always assume that k≠0; that is, the surgery on Kα/β is nontrivial. See [14] for the Conway notation of 2-bridge knots. Here α and β are coprime integers given by the continued fraction
(19)αβ=-2s1+12q1+⋯+1/(-2sn+1/2qn),
where α>0, -α<β<α, and α (resp., β) is odd (resp., even), and si,qi≥1 for i=1,…,n.Two equivalent surgery descriptions of the surgery manifoldKα/β(h/k), where α/β=[-2s1,2q1,…,-2sn,2qn].
(a)
(b)Since every 2-bridge knot admits a Conway representation with an even number of even parameters (see exercise 2.1.14 of [15, p. 26]), we have that our family of surgery manifolds Mn(ri/si;pi/qi;h/k) contains all closed manifolds obtained by (nontrivial) Dehn surgeries on 2-bridge knots. Recall that a 2-bridge knot Kα/β is nonhyperbolic if and only if α=1, in which case it is the torus knot of type (2,β) (see, e.g., [5]). Since the surgery on torus knot is well understood (see [9]), we restrict our attention to hyperbolic 2-bridge knots. Ochiai proved that such manifolds have Heegaard genus 2 (see [10]). The following also gives a different proof of the Ochiai result together with an explicit 2-generator 2-relator geometric presentation of the fundamental group.Theorem 3.
LetKα/β(γ), γ=h/k≠∞, be the closed orientable 3-manifold obtained by γ Dehn surgery on the hyperbolic 2-bridge knot Kα/β, where α/β=[-2s1,2q1,…,-2sn,2qn]. Then the fundamental group of Kα/β(γ) admits a geometric presentation with generators a1 and c and two relators deduced from the recurrence formulae:
(20)ai+1=ckbi-qic-kaibiqi,bi+1=ai+1-si+1bic-kai+1si+1ck
for i=1,…,n-1, where b1=a1-s1c-ka1s1ck. In particular, the surgery manifold Kα/β(γ) has Heegaard genus 2.Proof.
By Theorem1, the fundamental group of Kα/β(γ) has a presentation with generators ai, bi, and c, i=1,…,n, and relations
(21)ai-1=biqi⋯bnqnckbn-qn⋯bi-qic-k,bi-1=c-kai-si⋯a1-s1cka1s1⋯aisi,c-h=bn-qn⋯b1-q1a1s1b1q1⋯ansnbnqnan-sn⋯a1-s1.
This presentation is geometric; that is, it is induced by a genus2n+1 Heegaard diagram of Kα/β(γ). We can eliminate the generator b1-1=c-ka1-s1cka1s1 to get a balanced presentation of π1(Kα/β(γ)) with 2n generators. We see that the curve of the diagram represented by the relator b1c-ka1-s1cka1s1 has exactly one point in common with the curve (on the Heegaard surface) represented by the generator b1. Then the pair of such curves determines a reducible handle in the diagram. Cancelling it yields a new Heegaard diagram of Kα/β(γ) (with genus 2n) inducing the above 2n-balanced presentation for π1(Kα/β(γ)). The recurrence formulae of the statement are obtained as follows:
(22)ai+1-1=bi+1qi+1⋯bnqnckbn-qn⋯bi+1-qi+1c-k=bi-qibiqibi+1qi+1⋯bnqnckbn-qn⋯bi+1-qi+1bi-qibiqic-k=bi-qiai-1ckbiqic-k,bi+1-1=c-kai+1-si+1ai-si⋯a1-s1cka1s1⋯aisiai+1si+1=c-kai+1-si+1ckbi-1ai+1si+1
for i=1,…,n-1. Using these relations we can successively eliminate the generators ai+1 and bi+1 for i=1,…,n-1 (together with b1=a1-s1c-ka1s1ck). The Tietze moves on the obtained presentations for the group π1(Kα/β(γ)) correspond geometrically to cancel reducible handles in the current Heegaard diagrams (of decreasing genus) inducing those presentations. So Kα/β(γ) can be represented by a Heegaard diagram of genus 2. Such a diagram induces a geometric presentation for π1(Kα/β(γ)) with two generators a1 and c and two relators obtained by applying the above recurrence algorithm. This shows that the genus of Kα/β(γ) is at most 2. Now we claim that the genus is exactly 2. This follows from the fact that 2-bridge knots have tunnel number equal to one and no lens space surgeries (see, e.g., [5]).To complete the section we write explicitly the geometric presentations forπ1(Kα/β(γ)) with α/β=[-2s1,2q1,…,-2sn,2qn] for n=1,2.Corollary 4.
The fundamental group of the surgery manifoldKα/β(γ), γ=h/k and α/β=[-2s1,2q1]=(-4q1s1+1)/(2q1), has the geometric presentation:
(23)π1(Kα/β(γ))=〈a1,c:a1[a1-s1,c-k]q1[a1-s1,ck]q1=1,ch[c-k,a1-s1]q1[c-k,a1s1]q1=1〉,
where [x,y]=xyx-1y-1.Corollary 5.
The fundamental group of the surgery manifoldKα/β(γ), γ=h/k, α/β=[-2s1,2q1,-2s2,2q2], that is, α=16q1q2s1s2-4(q1s1+q2s1+q2s2)+1 and β=-8q1q2s2+2(q1+q2), has the geometric presentation with generators a1 and c and relations a2[b2q2,ck]=1 and
(24)chb2-q2b1-q1a1s1b1q1a2s2b2q2a2-s2a1-s1=1,
where
(25)b1=[a1-s1,c-k],a2=[a1-s1,ck]q1a1[a1-s1,c-k]q1,b2=([a1-s1,ck]q1a1[a1-s1,c-k]q1)-s2[a1-s1,c-k]c-k×([a1-s1,ck]q1a1[a1-s1,c-k]q1)s2ck.From Theorem3 and [5] we also have the following consequence (for n=1 see [6]).Corollary 6.
LetKα/β be a hyperbolic 2-bridge knot, where α/β=[-2s1,2q1,…,-2sn,2qn] and n≥2. Then the surgery manifolds Kα/β(γ), γ≠∞, are hyperbolic and have Heegaard genus 2. The volumes of such manifolds can be made arbitrarily large.Proof.
As done in [16, p. 725], for a slightly different link (see also [11, 17]), it follows that the links ℒ2n+1 are hyperbolic with volume approximately (2n-1)(7.32772…). Furthermore, ℒ2n+1 is amphicheiral and its symmetry group is isomorphic to ℤ2×D4, where D4 is the dihedral group of order 8. On choosing a framing for each unknotted component of ℒ2n+1, we can perform 1/n Dehn surgery on each of the unknotted components of ℒ2n+1. This produces the hyperbolic 2-bridge knot Kn=Kα/β, where α/β=[-2n,2n,…,-2n,2n]. Thurston’s hyperbolic Dehn surgery theorem [3] in this context says that Km has a 2n-long continued fraction consisting of 2m’s with volumes of 𝕊3∖Km converging to that of 𝕊3∖ℒ2n+1 as m goes to infinity. Since these are getting arbitrarily large, the result follows. In fact, the volumes of the surgery hyperbolic manifolds Kn(γ), γ≠∞ and n≥2, become arbitrarily large as n goes to infinity. The fact that the volumes of these manifolds can be arbitrarily large is also a consequence of work by Lackenby on volumes of hyperbolic alternating links (see [18]). (See, e.g., [19, 20] for interesting estimates of volumes for hyperbolic manifolds arising from right-angled Coxeter polyhedra.)


## 2. Covering Properties

In this section we study covering properties of our surgery manifolds. Using Montesinos’ trick [8], we prove that such manifolds are 2-fold branched covers of a connected sum of lens spaces. Moreover, it follows that a very large subclass of our surgery manifolds are 2-fold coverings of the 3-sphere branched over well-specified clearly depicted links. Finally, we show explicitly what the branched cover looks like for the surgeries on a large class of links including 2-bridge knots as very particular case.Theorem 7.
Suppose thatri is odd for every i=1,…,n. Then the surgery manifold Mn(ri/si;pi/qi;h/k) is 2-fold branched covering of the connected sum of n lens spaces L(r1,2s1)#⋯#L(rn,2sn).Proof.
As shown in Figure4(a), there is an orientation-preserving involution ρ in 𝕊3 which induces an involution with two fixed points (resp., without fixed points) in each component L0 and Ki (resp., Li) of ℒ2n+1, for i=1,…,n. Here we will assume n≥2. For n=1 see [6]. Let ℒ′ be the link consisting of those components of ℒ2n+1 for which the number of fixed points of ρ is different from two. Let p:𝕊3→𝕊3 be the 2-fold cyclic branched covering of the 3-sphere 𝕊3 defined by ρ. By Theorem 2 of [8] the manifold obtained by doing surgery on ℒ2n+1 is a 2-fold cyclic covering branched over a manifold obtained by doing surgery on p(ℒ′). But p(ℒ′) is a trivial link. Now the result follows from the fact that surgery on a trivial link produces a connected sum of lens spaces. This yields a representation of our surgery manifolds as branched coverings of a connected sum of lens spaces.The 2-symmetric planar projections of the linksℒ2n+1 and ℒn+1′=ℒ2n+1(1/s1,…,1/sn).
(a)
(b)Letℒ′n+1 be the oriented link in 𝕊3 with n+1 components (which we denote by L0′ and Ki for i=1,…,n) obtained from ℒ2n+1 by doing 1/si Dehn surgeries on its Li components, i=1,…,n. The link ℒn+1′ is strongly invertible (see Figure 4(b)); that is, there is an orientation-preserving involution of 𝕊3, also denoted by ρ, which induces in each component of ℒn+1′ an involution with two fixed points. We remark that in Figure 4(b) the last string has 2sn-1 crossings instead of 2sn (sn≥1) because we have shifted the subarc (at the final crossing) of the link from the bottom to the top. This permits losing a crossing. Now we recall the statement of Theorem 1 from [8]: let M be a closed orientable 3-manifold that is obtained by doing surgery on a strongly-invertible link L of n components. Then M is a 2-fold cyclic covering of the 3-sphere branched over a link of at most n+1 components. Thus Theorem 1 of [8] applies to our case, and we can state that the manifolds Mn(ri/si;pi/qi;h/k) with ri=1, i=1,…,n, are 2-fold coverings of 𝕊3 branched over a link of at most n+2 components. Now we apply the Montesinos algorithm, given in [8], to describe explicitly the branch sets of the above 2-fold branched coverings. Let ℒr(pi/qi;h/k), where r=2s1+⋯+2sn, denote the branch set of the 2-fold branched covering Mn(ri/si;pi/qi;h/k), with ri=1 for i=1,…,n, of 𝕊3 which corresponds to the involution ρ shown in Figure 4(b) (recall that si≥1 for i=1,…,n). Let mi=yi be the meridians of the components Ki of ℒn+1′ and m0=x0 the meridian of the component L0′ of ℒn+1′. The pair (mi,li), where li is the longitude of Ki, is a preferred frame; that is, li~0 in the exterior space 𝕊3∖Ki and lk(mi,li)=1 for i=1,…,n. The pair (m0,l0′), where l0′ is the longitude of L0′, is not a preferred frame since l0′~-(r-1)m0 in 𝕊3∖L0′, where r=2s1+⋯+2sn. To have a preferred frame, we take the pair (m0,l0), where l0=l0′+(r-1)m0. Let V be a regular neighbourhood of the link ℒn+1′ in 𝕊3. Without loss of generality, we can choose V, the meridians mi, and the longitudes li, i=0,1,…,n, to be invariant under the involution ρ. The quotient space of 𝕊3 under ρ is illustrated in Figure 5. The image of V under the projection π:𝕊3→𝕊3/ρ consists of n+1 disjoint 3-balls; B0,B1,…,Bn, say. To obtain the branch set ℒr(pi/qi;h/k), where r=2s1+⋯+2sn, via the Montesinos algorithm, we isotopy the Bi’s along the images π(li) of the longitudes li and replace them by an h/k rational tangle for i=0 and by pi/qi rational tangles, for i=1,…,n, as in Figure 6.Figure 5
The quotient(𝕊3∖intV)/ρ.Figure 6
The linkℒr(pi/qi;h/k), r=2s1+⋯+2sn.Summarizing, we have proven the following main result.Theorem 8.
Letℳ=Mn(ri/si;pi/qi;h/k), ri=1 and si≥1, for i=1,…,n, be the closed connected orientable 3-manifold obtained by Dehn surgeries on the components of the link ℒ2n+1. Then ℳ is the 2-fold covering of the 3-sphere branched over the link ℒr(pi/qi;h/k), where r=2s1+⋯+2sn, pictured in Figure 6.Theorem 9.
LetKα/β(γ), γ=h/k≠∞, be the closed connected orientable 3-manifold obtained by γ Dehn surgery on the hyperbolic 2-bridge knot Kα/β, where α/β=[-2s1,2q1,…,-2sn,2qn]. Then Kα/β(γ) is the 2-fold covering of the 3-sphere branched over the link ℒr(pi/qi;h/k), where r=2s1+⋯+2sn and pi=1 for every i=1,…,n.For example,Kα/β(γ), γ=h/k≠∞, where α/β=[-4,6,-6,8], hence α=1049, and β=-274, is the 2-fold covering of the 3-sphere 𝕊3 branched over the link L10(1/3,1/4;h/k) as shown in Figure 7.Figure 7
The linkℒ10(1/3,1/4;h/k).



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