Research Article | Open Access
Fundamental Group and Covering Properties of Hyperbolic Surgery Manifolds
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.
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  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., ). 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 , let be the oriented link with components , , and , , in the oriented 3-sphere 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 . Let us consider the closed connected orientable 3-manifolds obtained by Dehn surgery on along the oriented link such that the surgery coefficients , , and correspond to the oriented components , , and , respectively, where . Of course, we always assume that , , and . 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 ). 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 ). Now we determine a geometric presentation for the fundamental group of the surgery manifold . 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 has generators , , and , for every (see Figure 1).
The meridians and and the longitudes and of the components and , respectively, of are where and for every . The meridian and the longitude of the component of are
To determine the formulae for longitudes , , and , we have used the following procedure. Fix an orientation and an initial point for each component of the link . 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 manifold is obtained from that of by adding the relations for . Since the integers of the pairs , , and are coprime, there are integers , , , , , and such that Let us define for .
Then we have for . We have the following result.
Theorem 1. The fundamental group of the surgery 3-dimensional manifold admits the finite balanced presentation with generators , , and , , and relations:
The closed manifold admits a Heegaard diagram of genus inducing the above presentation, which is thus geometric. Furthermore, the Heegaard genus of is at most .
Proof. Substituting the above relations in the relators of the Wirtinger presentation of and using the previous formulae for the longitudes , , and , we get the relations of the statement. More precisely, substituting , , and into we get or, equivalently, which is the first relation of the statement for . Then we have or, equivalently, which is the second relation of the statement for . From the expression of we get or, equivalently, which is the first relation of the statement for . From the expression of we get or, equivalently, which is the second relation of the statement for . Going on like this, we get by finite iteration the first and second relations of the statement for . Substituting , , and into we get 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].
We also note that the first integral homology group of is isomorphic to . For example, if , , then the Heegaard genus of our surgery manifolds is exactly .
Theorem 2. For any integer and for almost all pairs of surgery coefficients , , and , the closed connected orientable 3-manifolds are hyperbolic.
If for every , then the surgery 3-dimensional manifold is homeomorphic to the closed orientable 3-manifold obtained by Dehn surgery on the 2-bridge knot corresponding to the Conway parameters , as shown in Figure 3. Note that our parameterization is coherent with that used by Rolfsen [4, p. 303], by setting , , and so on. The 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 and have opposite signs for odd. In particular, is negative for odd since . We always assume that ; that is, the surgery on is nontrivial. See  for the Conway notation of 2-bridge knots. Here and are coprime integers given by the continued fraction where , , and (resp., ) is odd (resp., even), and for .
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 contains all closed manifolds obtained by (nontrivial) Dehn surgeries on 2-bridge knots. Recall that a 2-bridge knot is nonhyperbolic if and only if , in which case it is the torus knot of type (see, e.g., ). Since the surgery on torus knot is well understood (see ), we restrict our attention to hyperbolic 2-bridge knots. Ochiai proved that such manifolds have Heegaard genus 2 (see ). 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. Let , , be the closed orientable 3-manifold obtained by Dehn surgery on the hyperbolic 2-bridge knot , where . Then the fundamental group of admits a geometric presentation with generators and and two relators deduced from the recurrence formulae: for , where . In particular, the surgery manifold has Heegaard genus 2.
Proof. By Theorem 1, the fundamental group of has a presentation with generators , , and , , and relations
This presentation is geometric; that is, it is induced by a genus Heegaard diagram of . We can eliminate the generator to get a balanced presentation of with generators. We see that the curve of the diagram represented by the relator has exactly one point in common with the curve (on the Heegaard surface) represented by the generator . Then the pair of such curves determines a reducible handle in the diagram. Cancelling it yields a new Heegaard diagram of (with genus ) inducing the above -balanced presentation for . The recurrence formulae of the statement are obtained as follows: for . Using these relations we can successively eliminate the generators and for (together with ). The Tietze moves on the obtained presentations for the group correspond geometrically to cancel reducible handles in the current Heegaard diagrams (of decreasing genus) inducing those presentations. So can be represented by a Heegaard diagram of genus 2. Such a diagram induces a geometric presentation for with two generators and and two relators obtained by applying the above recurrence algorithm. This shows that the genus of 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., ).
To complete the section we write explicitly the geometric presentations for with for .
Corollary 4. The fundamental group of the surgery manifold , and , has the geometric presentation: where .
Corollary 5. The fundamental group of the surgery manifold , , , that is, and , has the geometric presentation with generators and and relations and where
Corollary 6. Let be a hyperbolic 2-bridge knot, where and . Then the surgery manifolds , , 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 are hyperbolic with volume approximately . Furthermore, is amphicheiral and its symmetry group is isomorphic to , where is the dihedral group of order 8. On choosing a framing for each unknotted component of , we can perform Dehn surgery on each of the unknotted components of . This produces the hyperbolic 2-bridge knot , where . Thurston’s hyperbolic Dehn surgery theorem  in this context says that has a -long continued fraction consisting of ’s with volumes of converging to that of as goes to infinity. Since these are getting arbitrarily large, the result follows. In fact, the volumes of the surgery hyperbolic manifolds , and , become arbitrarily large as 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 ). (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 , 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 that is odd for every . Then the surgery manifold is 2-fold branched covering of the connected sum of lens spaces .
Proof. As shown in Figure 4(a), there is an orientation-preserving involution in which induces an involution with two fixed points (resp., without fixed points) in each component and (resp., ) of , for . Here we will assume . For see . Let be the link consisting of those components of for which the number of fixed points of is different from two. Let be the 2-fold cyclic branched covering of the 3-sphere defined by . By Theorem 2 of  the manifold obtained by doing surgery on is a 2-fold cyclic covering branched over a manifold obtained by doing surgery on . But 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.
Let be the oriented link in with components (which we denote by and for ) obtained from by doing Dehn surgeries on its components, . The link is strongly invertible (see Figure 4(b)); that is, there is an orientation-preserving involution of , also denoted by , which induces in each component of an involution with two fixed points. We remark that in Figure 4(b) the last string has crossings instead of () 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 : let be a closed orientable 3-manifold that is obtained by doing surgery on a strongly-invertible link of components. Then is a 2-fold cyclic covering of the 3-sphere branched over a link of at most components. Thus Theorem 1 of  applies to our case, and we can state that the manifolds with , , are 2-fold coverings of branched over a link of at most components. Now we apply the Montesinos algorithm, given in , to describe explicitly the branch sets of the above 2-fold branched coverings. Let , where , denote the branch set of the 2-fold branched covering , with for , of which corresponds to the involution shown in Figure 4(b) (recall that for ). Let be the meridians of the components of and the meridian of the component of . The pair , where is the longitude of , is a preferred frame; that is, in the exterior space and for . The pair , where is the longitude of , is not a preferred frame since in , where . To have a preferred frame, we take the pair , where . Let be a regular neighbourhood of the link in . Without loss of generality, we can choose , the meridians , and the longitudes , , to be invariant under the involution . The quotient space of under is illustrated in Figure 5. The image of under the projection consists of disjoint 3-balls; , say. To obtain the branch set , where , via the Montesinos algorithm, we isotopy the ’s along the images of the longitudes and replace them by an rational tangle for and by rational tangles, for , as in Figure 6.
Summarizing, we have proven the following main result.
Theorem 8. Let , and , for , be the closed connected orientable 3-manifold obtained by Dehn surgeries on the components of the link . Then is the 2-fold covering of the 3-sphere branched over the link , where , pictured in Figure 6.
Theorem 9. Let , , be the closed connected orientable 3-manifold obtained by Dehn surgery on the hyperbolic 2-bridge knot , where . Then is the 2-fold covering of the 3-sphere branched over the link , where and for every .
For example, , , where , hence , and , is the 2-fold covering of the 3-sphere branched over the link as shown in Figure 7.
This work is performed under the auspices of the GNSAGA of the National Research Council (CNR) of Italy and partially supported by the Ministero dell’Istruzione, dell’Universitá e della Ricerca Scientifica (MIUR) of Italy.
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