Fundamental Group and Covering Properties of Hyperbolic Surgery Manifolds

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.

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 ℒ_2𝑛+1 be the oriented link with 2𝑛+1 components 𝐿₀, 𝐿_𝑖, and 𝐾_𝑖, 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 𝑀_𝑛(𝑟_𝑖/𝑠_𝑖;𝑝_𝑖/𝑞_𝑖;ℎ/𝑘) obtained by Dehn surgery on 𝕊³ along the oriented link ℒ_2𝑛+1 such that the surgery coefficients 𝑟_𝑖/𝑠_𝑖, 𝑝_𝑖/𝑞_𝑖, and ℎ/𝑘 correspond to the oriented components 𝐿_𝑖, 𝐾_𝑖, and 𝐿₀, respectively, where 𝑖=1,…,𝑛. Of course, we always assume that gcd(𝑟_𝑖,𝑠_𝑖)=1, gcd(𝑝_𝑖,𝑞_𝑖)=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 𝑀_𝑛(𝑟_𝑖/𝑠_𝑖;𝑝_𝑖/𝑞_𝑖;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(ℒ_2𝑛+1)=𝜋₁(𝕊³⧵ℒ_2𝑛+1) has generators 𝑥₀, 𝑥_𝑖, and 𝑦_𝑖, for every 𝑖=1,…,𝑛 (see Figure 1).

Figure 1 

Dehn surgery description of the 3-manifold 𝑀𝑛(𝑟𝑖/𝑠𝑖;𝑝𝑖/𝑞𝑖;ℎ/𝑘) and the generators of a Wirtinger presentation of 𝜋(ℒ2𝑛+1).

The meridians 𝐦_𝑖 and 𝝁_𝑖 and the longitudes 𝓵_𝑖 and 𝝀_𝑖 of the components 𝐿_𝑖 and 𝐾_𝑖, respectively, of ℒ_2𝑛+1 are 𝐦_𝑖=𝑥_𝑖,𝓵_𝑖=𝑦⁻¹_𝑖−1𝑥⁻¹_𝑖−1⋯𝑦₁⁻¹𝑥₁⁻¹𝑦₁⋯𝑦_𝑛𝑥₀𝑦_𝑛⁻¹⋯𝑦₁⁻¹𝑥₁𝑦₁⋯𝑥_𝑖−1𝑦_𝑖−1𝑥⁻¹_𝑖−1⋯𝑥₁⁻¹𝑥₀⁻¹𝑥₁⋯𝑥_𝑖−1,𝝁_𝑖=𝑦_𝑖,𝝀_𝑖=𝑦_𝑖⋯𝑦_𝑛𝑥₀⁻¹𝑦_𝑛⁻¹⋯𝑦_𝑖⁻¹𝑥_𝑖⁻¹𝑦⁻¹_𝑖−1⋯𝑥₂⁻¹𝑦₁⁻¹𝑥₁⁻¹𝑦₁⋯𝑦_𝑛𝑥₀𝑦_𝑛⁻¹⋯𝑦₁⁻¹𝑥₁𝑦₁𝑥₂⋯𝑦_𝑖−1𝑥_𝑖, (1) where [𝐦_𝑖,𝓵_𝑖]=1 and [𝝁_𝑖,𝝀_𝑖]=1 for every 𝑖=1,…,𝑛. The meridian 𝐦₀ and the longitude 𝓵₀ of the component 𝐿₀ of ℒ_2𝑛+1 are 𝐦₀=𝑥₀,𝓵₀=𝑦_𝑛⁻¹⋯𝑦₁⁻¹𝑥₁𝑦₁⋯𝑥_𝑛𝑦_𝑛𝑥_𝑛⁻¹⋯𝑥₁⁻¹. (2)

To determine the formulae for longitudes 𝓵_𝑖, 𝝀_𝑖, and 𝓵₀, we have used the following procedure. Fix an orientation and an initial point for each component of the link ℒ_2𝑛+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 manifold 𝑀_𝑛(𝑟_𝑖/𝑠_𝑖;𝑝_𝑖/𝑞_𝑖;ℎ/𝑘) is obtained from that of 𝜋(ℒ_2𝑛+1) by adding the relations 𝐦^𝑟𝑖_𝑖𝓵^𝑠𝑖_𝑖=1,𝝁^𝑝𝑖_𝑖𝝀^𝑞𝑖_𝑖=1,𝐦^ℎ₀𝓵^𝑘₀=1, (3) for 𝑖=1,…,𝑛. Since the integers of the pairs (𝑝_𝑖,𝑞_𝑖), (𝑟_𝑖,𝑠_𝑖), and (ℎ,𝑘) are coprime, there are integers 𝛼_𝑖, 𝛽_𝑖, 𝛾_𝑖, 𝛿_𝑖, 𝜉, and 𝜂 such that 𝑞_𝑖𝛼_𝑖−𝑝_𝑖𝛽_𝑖𝑠=1,_𝑖𝛾_𝑖−𝑟_𝑖𝛿_𝑖=1,𝑘𝜉−ℎ𝜂=1. (4) Let us define 𝑎_𝑖∶=𝐦^𝛾𝑖_𝑖𝓵^𝛿𝑖_𝑖,𝑏_𝑖∶=𝝁^𝛼𝑖_𝑖𝝀^𝛽𝑖_𝑖,𝑐∶=𝐦^𝜉₀𝓵^𝜂₀, (5) for 𝑖=1,…,𝑛.

Then we have 𝑎^𝑠𝑖_𝑖=𝐦^𝛾𝑖_𝑖𝓵^𝛿𝑖_𝑖^𝑠𝑖=𝐦_𝑖𝐦^{𝑟𝑖𝛿𝑖}_𝑖𝓵^{𝛿𝑖𝑠𝑖}_𝑖=𝐦_𝑖𝐦^𝑟𝑖_𝑖𝓵^𝑠𝑖_𝑖^𝛿𝑖=𝐦_𝑖=𝑥_𝑖,𝑏^𝑞𝑖_𝑖=𝝁^𝛼𝑖_𝑖𝝀^𝛽𝑖_𝑖^𝑞𝑖=𝝁_𝑖𝝁^{𝑝𝑖𝛽𝑖}_𝑖𝝀^{𝑞𝑖𝛽𝑖}_𝑖=𝝁_𝑖𝝁^𝑝𝑖_𝑖𝝀^𝑞𝑖_𝑖^𝛽𝑖=𝝁_𝑖=𝑦_𝑖,𝑐^𝑘=𝐦^𝜉₀𝓵^𝜂₀^𝑘=𝐦₀𝐦₀^ℎ𝜂𝓵₀^𝑘𝜂=𝐦₀𝐦^ℎ₀𝓵^𝑘₀^𝜂=𝐦₀=𝑥₀,𝑎^−𝑟𝑖_𝑖=𝐦^𝛾𝑖_𝑖𝓵^𝛿𝑖_𝑖^−𝑟𝑖=𝐦^{−𝑟𝑖𝛾𝑖}_𝑖𝓵^{−𝑠𝑖𝛾𝑖}_𝑖𝓵_𝑖=𝐦^𝑟𝑖_𝑖𝓵^𝑠𝑖_𝑖^−𝛾𝑖𝓵_𝑖=𝓵_𝑖,𝑏^−𝑝𝑖_𝑖=𝝁^𝛼𝑖_𝑖𝝀^𝛽𝑖_𝑖^−𝑝𝑖=𝝁^{−𝑝𝑖𝛼𝑖}_𝑖𝝀^{−𝑞𝑖𝛼𝑖}_𝑖𝝀_𝑖=𝝁^𝑝𝑖_𝑖𝝀^𝑞𝑖_𝑖^−𝛼𝑖𝝀_𝑖=𝝀_𝑖,𝑐^−ℎ=𝐦^𝜉₀𝓵^𝜂₀^−ℎ=𝐦₀^−ℎ𝜉𝓵₀^−𝑘𝜉𝓵₀=𝐦^ℎ₀𝓵^𝑘₀^−𝜉𝓵₀=𝓵₀ (6) for 𝑖=1,…,𝑛. We have the following result.

Theorem 1. The fundamental group of the surgery 3-dimensional manifold 𝑀_𝑛(𝑟_𝑖/𝑠_𝑖;𝑝_𝑖/𝑞_𝑖;ℎ/𝑘) admits the finite balanced presentation with 2𝑛+1 generators 𝑎_𝑖, 𝑏_𝑖, and 𝑐, 𝑖=1,…,𝑛, and 2𝑛+1 relations: 𝑎^𝑟𝑖_𝑖𝑏^𝑞𝑖_𝑖⋯𝑏^𝑞𝑛_𝑛𝑐^𝑘𝑏^−𝑞𝑛_𝑛⋯𝑏^−𝑞𝑖_𝑖𝑐^−𝑘𝑏=1,_𝑝𝑖𝑖𝑐^−𝑘𝑎^−𝑠𝑖_𝑖⋯𝑎^−𝑠1₁𝑐^𝑘𝑎^𝑠1₁⋯𝑎^𝑠𝑖_𝑖𝑐=1,_ℎ𝑏^−𝑞𝑛_𝑛⋯𝑏^−𝑞1₁𝑎^𝑠1₁𝑏^𝑞1₁⋯𝑎^𝑠𝑛_𝑛𝑏^𝑞𝑛_𝑛𝑎^−𝑠𝑛_𝑛⋯𝑎^−𝑠1₁=1. (7)
The closed manifold 𝑀_𝑛(𝑟_𝑖/𝑠_𝑖;𝑝_𝑖/𝑞_𝑖;ℎ/𝑘) admits a Heegaard diagram of genus 2𝑛+1 inducing the above presentation, which is thus geometric. Furthermore, the Heegaard genus of 𝑀_𝑛(𝑟_𝑖/𝑠_𝑖;𝑝_𝑖/𝑞_𝑖;ℎ/𝑘) is at most 2𝑛+1.

Proof. Substituting the above relations in the relators of the Wirtinger presentation of 𝜋(ℒ_2𝑛+1) and using the previous formulae for the longitudes 𝓵_𝑖, 𝝀_𝑖, and 𝓵₀, we get the relations of the statement. More precisely, substituting 𝓵₁=𝑎^−𝑟1₁, 𝑦_𝑖=𝑏^𝑞𝑖_𝑖, and 𝑥₀=𝑐^𝑘 into 𝓵₁=𝑦₁⋯𝑦_𝑛𝑥₀𝑦_𝑛⁻¹⋯𝑦₁⁻¹𝑥₀⁻¹ (8) we get 𝑎^−𝑟1₁=𝑏^𝑞1₁⋯𝑏^𝑞𝑛_𝑛𝑐^𝑘𝑏^−𝑞𝑛_𝑛⋯𝑏^−𝑞1₁𝑐^−𝑘 (9) or, equivalently, 𝑎^𝑟1₁𝑏^𝑞1₁⋯𝑏^𝑞𝑛_𝑛𝑐^𝑘𝑏^−𝑞𝑛_𝑛⋯𝑏^−𝑞1₁𝑐^−𝑘=1 (10) which is the first relation of the statement for 𝑖=1. Then we have 𝑏^−𝑝1₁=𝝀₁=𝑦₁⋯𝑦_𝑛𝑥₀⁻¹𝑦_𝑛⁻¹⋯𝑦₁⁻¹𝑥₁⁻¹×𝑦₁⋯𝑦_𝑛𝑥₀𝑦_𝑛⁻¹⋯𝑦₁⁻¹𝑥₁=𝓵₁𝑥₀⁻¹𝑥₁⁻¹𝓵₁𝑥₀𝑥₁=𝑐^−𝑘𝑎^𝑟1𝑎^−𝑠1₁𝑎^−𝑟1₁𝑐^𝑘𝑎^𝑠1₁=𝑐^−𝑘𝑎^−𝑠1₁𝑐^𝑘𝑎^𝑠1₁ (11) or, equivalently, 𝑏^𝑝1₁𝑐^−𝑘𝑎^−𝑠1₁𝑐^𝑘𝑎^𝑠1₁=1 (12) which is the second relation of the statement for 𝑖=1. From the expression of 𝓵₂ we get 𝑎^−𝑟2₂=𝓵₂=𝑦₁⁻¹𝑥₁⁻¹𝑦₁⋯𝑦_𝑛𝑥₀𝑦_𝑛⁻¹⋯𝑦₁⁻¹𝑥₁𝑦₁𝑥₁⁻¹𝑥₀⁻¹𝑥₁=𝑏^−𝑞1₁𝑎^−𝑠1₁𝓵₁𝑥₀𝑎^𝑠1₁𝑏^𝑞1₁𝑎^−𝑠1₁𝑐^−𝑘𝑎^𝑠1₁=𝑏^−𝑞1₁𝑎^−𝑠1₁𝑎^−𝑟1₁𝑐^𝑘𝑎^𝑠1₁𝑏^𝑞1₁𝑎^−𝑠1₁𝑐^−𝑘𝑎^𝑠1₁=𝑏^−𝑞1₁𝑎^−𝑟1₁𝑎^−𝑠1₁𝑐^𝑘𝑎^𝑠1₁𝑏^𝑞1₁𝑎^−𝑠1₁𝑐^−𝑘𝑎^𝑠1₁=𝑏^−𝑞1₁𝑎^−𝑟1₁𝑐^𝑘𝑏^−𝑝1₁𝑏^𝑞1₁𝑏^𝑝1₁𝑐^−𝑘=𝑏^−𝑞1₁𝑎^−𝑟1₁𝑐^𝑘𝑏^𝑞1₁𝑐^−𝑘=𝑏^−𝑞1₁𝑏^𝑞1₁⋯𝑏^𝑞𝑛_𝑛𝑐^𝑘𝑏^−𝑞𝑛_𝑛⋯𝑏^−𝑞1₁𝑐^−𝑘𝑐^𝑘𝑏^𝑞1₁𝑐^−𝑘=𝑏^𝑞2₂⋯𝑏^𝑞𝑛_𝑛𝑐^𝑘𝑏^−𝑞𝑛_𝑛⋯𝑏^−𝑞2₂𝑐^−𝑘 (13) or, equivalently, 𝑎^𝑟2₂𝑏^𝑞2₂⋯𝑏^𝑞𝑛_𝑛𝑐^𝑘𝑏^−𝑞𝑛_𝑛⋯𝑏^−𝑞2₂𝑐^−𝑘=1 (14) which is the first relation of the statement for 𝑖=2. From the expression of 𝝀₂ we get 𝑏^−𝑝2₂=𝝀₂=𝑦₂⋯𝑦_𝑛𝑥₀⁻¹𝑦_𝑛⁻¹⋯𝑦₂⁻¹𝑥₂⁻¹𝑦₁⁻¹𝑥₁⁻¹×𝑦₁⋯𝑦_𝑛𝑥₀𝑦_𝑛⁻¹⋯𝑦₁⁻¹𝑥₁𝑦₁𝑥₂=𝑏^𝑞2₂⋯𝑏^𝑞𝑛_𝑛𝑐^−𝑘𝑏^−𝑞𝑛_𝑛⋯𝑏^−𝑞2₂𝑎^−𝑠2₂𝑏^−𝑞1₁𝑎^−𝑠1₁×𝑏^𝑞1₁⋯𝑏^𝑞𝑛_𝑛𝑐^𝑘𝑏^−𝑞𝑛_𝑛⋯𝑏^−𝑞1₁𝑎^𝑠1₁𝑏^𝑞1₁𝑎^𝑠2₂=𝑐^−𝑘𝑎^𝑟2₂𝑎^−𝑠2₂𝑏^−𝑞1₁𝑎^−𝑠1₁𝑎^−𝑟1₁𝑐^𝑘𝑎^𝑠1₁𝑏^𝑞1₁𝑎^𝑠2₂=𝑐^−𝑘𝑎^−𝑠2₂𝑎^𝑟2₂𝑎^−𝑟2₂𝑎^−𝑠1₁𝑐^𝑘𝑎^𝑠1₁𝑎^𝑠2₂=𝑐^−𝑘𝑎^−𝑠2₂𝑎^−𝑠1₁𝑐^𝑘𝑎^𝑠1₁𝑎^𝑠2₂ (15) or, equivalently, 𝑏^𝑝2₂𝑐^−𝑘𝑎^−𝑠2₂𝑎^−𝑠1₁𝑐^𝑘𝑎^𝑠1₁𝑎^𝑠2₂=1 (16) 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, 𝑦_𝑖=𝑏^𝑞𝑖_𝑖, and 𝑥_𝑖=𝑎^𝑠𝑖_𝑖 into 𝓵₀=𝑦_𝑛⁻¹⋯𝑦₁⁻¹𝑥₁𝑦₁⋯𝑥_𝑛𝑦_𝑛𝑥_𝑛⁻¹⋯𝑥₁⁻¹ (17) we get 𝑐^−ℎ=𝑏^−𝑞𝑛_𝑛⋯𝑏^−𝑞1₁𝑎^𝑠1₁𝑏^𝑞1₁⋯𝑎^𝑠𝑛_𝑛𝑏^𝑞𝑛_𝑛𝑎^−𝑠𝑛_𝑛⋯𝑎^−𝑠1₁ (18) 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 genus 2𝑛+1 inducing the presentation of 𝜋1(𝑀𝑛(𝑟𝑖/𝑠𝑖;𝑝𝑖/𝑞𝑖;ℎ/𝑘)).

We also note that the first integral homology group of 𝑀_𝑛(𝑟_𝑖/𝑠_𝑖;𝑝_𝑖/𝑞_𝑖;ℎ/𝑘) is isomorphic to ⨁^𝑛_𝑖=1(ℤ_|𝑟𝑖|⊕ℤ_|𝑝𝑖|)⊕ℤ_|ℎ|. For example, if 𝑟_𝑖=𝑝_𝑖=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 ℒ_2𝑛+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 integer 𝑛≥1 and for almost all pairs of surgery coefficients 𝑟_𝑖/𝑠_𝑖, 𝑝_𝑖/𝑞_𝑖, and ℎ/𝑘, the closed connected orientable 3-manifolds 𝑀_𝑛(𝑟_𝑖/𝑠_𝑖;𝑝_𝑖/𝑞_𝑖;ℎ/𝑘) are hyperbolic.

If 𝑟_𝑖=𝑝_𝑖=1 for every 𝑖=1,…,𝑛, then the surgery 3-dimensional manifold 𝑀_𝑛(1/𝑠_𝑖;1/𝑞_𝑖;ℎ/𝑘) is homeomorphic to the closed orientable 3-manifold 𝐾_𝛼/𝛽(ℎ/𝑘) obtained by ℎ/𝑘 Dehn surgery on the 2-bridge knot 𝐾_𝛼/𝛽 corresponding to the Conway parameters [−2𝑠₁,2𝑞₁,…,−2𝑠_𝑛,2𝑞_𝑛], 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 𝑐_𝑖 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 𝑐_𝑖 and 2𝑠_𝑖 have opposite signs for 𝑖 odd. In particular, 𝑐_𝑖 is negative for 𝑖 odd since 𝑠_𝑖≥1. We always assume that 𝑘≠0; that is, the surgery on 𝐾_𝛼/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 𝛼𝛽=−2𝑠₁+12𝑞₁+⋯+1/−2𝑠_𝑛+1/2𝑞_𝑛, (19) where 𝛼>0, −𝛼<𝛽<𝛼, and 𝛼 (resp., 𝛽) is odd (resp., even), and 𝑠_𝑖,𝑞_𝑖≥1 for 𝑖=1,…,𝑛.

(a)

(b)

Figure 3 

Two equivalent surgery descriptions of the surgery manifold 𝐾𝛼/𝛽(ℎ/𝑘), where 𝛼/𝛽=[−2𝑠1,2𝑞1,…,−2𝑠𝑛,2𝑞𝑛].

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 𝑀_𝑛(𝑟_𝑖/𝑠_𝑖;𝑝_𝑖/𝑞_𝑖;ℎ/𝑘) contains all closed manifolds obtained by (nontrivial) Dehn surgeries on 2-bridge knots. Recall that a 2-bridge knot 𝐾_𝛼/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. Let 𝐾_𝛼/𝛽(𝛾), 𝛾=ℎ/𝑘≠∞, be the closed orientable 3-manifold obtained by 𝛾 Dehn surgery on the hyperbolic 2-bridge knot 𝐾_𝛼/𝛽, where 𝛼/𝛽=[−2𝑠₁,2𝑞₁,…,−2𝑠_𝑛,2𝑞_𝑛]. Then the fundamental group of 𝐾_𝛼/𝛽(𝛾) admits a geometric presentation with generators 𝑎₁ and 𝑐 and two relators deduced from the recurrence formulae: 𝑎_𝑖+1=𝑐^𝑘𝑏^−𝑞𝑖_𝑖𝑐^−𝑘𝑎_𝑖𝑏^𝑞𝑖_𝑖,𝑏_𝑖+1=𝑎^{−𝑠𝑖+1}_𝑖+1𝑏_𝑖𝑐^−𝑘𝑎^𝑠𝑖+1_𝑖+1𝑐^𝑘 (20) for 𝑖=1,…,𝑛−1, where 𝑏₁=𝑎^−𝑠1₁𝑐^−𝑘𝑎^𝑠1₁𝑐^𝑘. In particular, the surgery manifold 𝐾_𝛼/𝛽(𝛾) has Heegaard genus 2.

Proof. By Theorem 1, the fundamental group of 𝐾_𝛼/𝛽(𝛾) has a presentation with generators 𝑎_𝑖, 𝑏_𝑖, and 𝑐, 𝑖=1,…,𝑛, and relations 𝑎_𝑖⁻¹=𝑏^𝑞𝑖_𝑖⋯𝑏^𝑞𝑛_𝑛𝑐^𝑘𝑏^−𝑞𝑛_𝑛⋯𝑏^−𝑞𝑖_𝑖𝑐^−𝑘,𝑏_𝑖⁻¹=𝑐^−𝑘𝑎^−𝑠𝑖_𝑖⋯𝑎^−𝑠1₁𝑐^𝑘𝑎^𝑠1₁⋯𝑎^𝑠𝑖_𝑖,𝑐^−ℎ=𝑏^−𝑞𝑛_𝑛⋯𝑏^−𝑞1₁𝑎^𝑠1₁𝑏^𝑞1₁⋯𝑎^𝑠𝑛_𝑛𝑏^𝑞𝑛_𝑛𝑎^−𝑠𝑛_𝑛⋯𝑎^−𝑠1₁. (21)
This presentation is geometric; that is, it is induced by a genus 2𝑛+1 Heegaard diagram of 𝐾_𝛼/𝛽(𝛾). We can eliminate the generator 𝑏₁⁻¹=𝑐^−𝑘𝑎^−𝑠1₁𝑐^𝑘𝑎^𝑠1₁ to get a balanced presentation of 𝜋₁(𝐾_𝛼/𝛽(𝛾)) with 2𝑛 generators. We see that the curve of the diagram represented by the relator 𝑏₁𝑐^−𝑘𝑎^−𝑠1₁𝑐^𝑘𝑎^𝑠1₁ 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 𝐾_𝛼/𝛽(𝛾) (with genus 2𝑛) inducing the above 2𝑛-balanced presentation for 𝜋₁(𝐾_𝛼/𝛽(𝛾)). The recurrence formulae of the statement are obtained as follows: 𝑎⁻¹_𝑖+1=𝑏^𝑞𝑖+1_𝑖+1⋯𝑏^𝑞𝑛_𝑛𝑐^𝑘𝑏^−𝑞𝑛_𝑛⋯𝑏^{−𝑞𝑖+1}_𝑖+1𝑐^−𝑘=𝑏^−𝑞𝑖_𝑖𝑏^𝑞𝑖_𝑖𝑏^𝑞𝑖+1_𝑖+1⋯𝑏^𝑞𝑛_𝑛𝑐^𝑘𝑏^−𝑞𝑛_𝑛⋯𝑏^{−𝑞𝑖+1}_𝑖+1𝑏^−𝑞𝑖_𝑖𝑏^𝑞𝑖_𝑖𝑐^−𝑘=𝑏^−𝑞𝑖_𝑖𝑎_𝑖⁻¹𝑐^𝑘𝑏^𝑞𝑖_𝑖𝑐^−𝑘,𝑏⁻¹_𝑖+1=𝑐^−𝑘𝑎^{−𝑠𝑖+1}_𝑖+1𝑎^−𝑠𝑖_𝑖⋯𝑎^−𝑠1₁𝑐^𝑘𝑎^𝑠1₁⋯𝑎^𝑠𝑖_𝑖𝑎^𝑠𝑖+1_𝑖+1=𝑐^−𝑘𝑎^{−𝑠𝑖+1}_𝑖+1𝑐^𝑘𝑏_𝑖⁻¹𝑎^𝑠𝑖+1_𝑖+1 (22) for 𝑖=1,…,𝑛−1. Using these relations we can successively eliminate the generators 𝑎_𝑖+1 and 𝑏_𝑖+1 for 𝑖=1,…,𝑛−1 (together with 𝑏₁=𝑎^−𝑠1₁𝑐^−𝑘𝑎^𝑠1₁𝑐^𝑘). The Tietze moves on the obtained presentations for the group 𝜋₁(𝐾_𝛼/𝛽(𝛾)) correspond geometrically to cancel reducible handles in the current Heegaard diagrams (of decreasing genus) inducing those presentations. So 𝐾_𝛼/𝛽(𝛾) can be represented by a Heegaard diagram of genus 2. Such a diagram induces a geometric presentation for 𝜋₁(𝐾_𝛼/𝛽(𝛾)) with two generators 𝑎₁ and 𝑐 and two relators obtained by applying the above recurrence algorithm. This shows that the genus of 𝐾_𝛼/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 for 𝜋₁(𝐾_𝛼/𝛽(𝛾)) with 𝛼/𝛽=[−2𝑠₁,2𝑞₁,…,−2𝑠_𝑛,2𝑞_𝑛] for 𝑛=1,2.

Corollary 4. The fundamental group of the surgery manifold 𝐾_𝛼/𝛽(𝛾), 𝛾=ℎ/𝑘 and 𝛼/𝛽=[−2𝑠₁,2𝑞₁]=(−4𝑞₁𝑠₁+1)/(2𝑞₁), has the geometric presentation: 𝜋₁𝐾_𝛼/𝛽𝛾=𝑎₁,𝑐∶𝑎₁𝑎^−𝑠1₁,𝑐^−𝑘𝑞1𝑎^−𝑠1₁,𝑐^𝑘𝑞1𝑐=1,_ℎ𝑐^−𝑘,𝑎^−𝑠1₁^𝑞1𝑐^−𝑘,𝑎^𝑠1₁^𝑞1=1, (23) where [𝑥,𝑦]=𝑥𝑦𝑥⁻¹𝑦⁻¹.

Corollary 5. The fundamental group of the surgery manifold 𝐾_𝛼/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[𝑏^𝑞2₂,𝑐^𝑘]=1 and 𝑐^ℎ𝑏^−𝑞2₂𝑏^−𝑞1₁𝑎^𝑠1₁𝑏^𝑞1₁𝑎^𝑠2₂𝑏^𝑞2₂𝑎^−𝑠2₂𝑎^−𝑠1₁=1, (24) where 𝑏₁=𝑎^−𝑠1₁,𝑐^−𝑘,𝑎₂=𝑎^−𝑠1₁,𝑐^𝑘𝑞1𝑎₁𝑎^−𝑠1₁,𝑐^−𝑘𝑞1,𝑏₂=𝑎^−𝑠1₁,𝑐^𝑘𝑞1𝑎₁𝑎^−𝑠1₁,𝑐^{−𝑘𝑞1−𝑠2}𝑎^−𝑠1₁,𝑐^−𝑘𝑐^−𝑘×𝑎^−𝑠1₁,𝑐^𝑘𝑞1𝑎₁𝑎^−𝑠1₁,𝑐^{−𝑘𝑞1𝑠2}𝑐^𝑘. (25)

From Theorem 3 and [5] we also have the following consequence (for 𝑛=1 see [6]).

Corollary 6. Let 𝐾_𝛼/𝛽 be a hyperbolic 2-bridge knot, where 𝛼/𝛽=[−2𝑠₁,2𝑞₁,…,−2𝑠_𝑛,2𝑞_𝑛] and 𝑛≥2. Then the surgery manifolds 𝐾_𝛼/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 ℒ_2𝑛+1 are hyperbolic with volume approximately (2𝑛−1)(7.32772…). Furthermore, ℒ_2𝑛+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 ℒ_2𝑛+1, we can perform 1/𝑛 Dehn surgery on each of the unknotted components of ℒ_2𝑛+1. This produces the hyperbolic 2-bridge knot 𝐊_𝑛=𝐾_𝛼/𝛽, where 𝛼/𝛽=[−2𝑛,2𝑛,…,−2𝑛,2𝑛]. Thurston’s hyperbolic Dehn surgery theorem [3] in this context says that 𝐊_𝑚 has a 2𝑛-long continued fraction consisting of 2𝑚’s with volumes of 𝕊³⧵𝐊_𝑚 converging to that of 𝕊³⧵ℒ_2𝑛+1 as 𝑚 goes to infinity. Since these are getting arbitrarily large, the result follows. In fact, the volumes of the surgery hyperbolic manifolds 𝐊_𝑛(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 that 𝑟_𝑖 is odd for every 𝑖=1,…,𝑛. Then the surgery manifold 𝑀_𝑛(𝑟_𝑖/𝑠_𝑖;𝑝_𝑖/𝑞_𝑖;ℎ/𝑘) is 2-fold branched covering of the connected sum of 𝑛 lens spaces 𝐿(𝑟₁,2𝑠₁)#⋯#𝐿(𝑟_𝑛,2𝑠_𝑛).

Proof. As shown in Figure 4(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 𝐾_𝑖 (resp., 𝐿_𝑖) of ℒ_2𝑛+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 ℒ_2𝑛+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 ℒ_2𝑛+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.

(a)

(b)

Figure 4 

The 2-symmetric planar projections of the links ℒ2𝑛+1 and ℒ′𝑛+1=ℒ2𝑛+1(1/𝑠1,…,1/𝑠𝑛).

Let ℒ^′_𝑛+1 be the oriented link in 𝕊³ with 𝑛+1 components (which we denote by 𝐿^′₀ and 𝐾_𝑖 for 𝑖=1,…,𝑛) obtained from ℒ_2𝑛+1 by doing 1/𝑠_𝑖 Dehn surgeries on its 𝐿_𝑖 components, 𝑖=1,…,𝑛. The link ℒ^′_𝑛+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 ℒ^′_𝑛+1 an involution with two fixed points. We remark that in Figure 4(b) the last string has 2𝑠_𝑛−1 crossings instead of 2𝑠_𝑛 (𝑠_𝑛≥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 𝑀_𝑛(𝑟_𝑖/𝑠_𝑖;𝑝_𝑖/𝑞_𝑖;ℎ/𝑘) with 𝑟_𝑖=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 ℒ_𝑟(𝑝_𝑖/𝑞_𝑖;ℎ/𝑘), where 𝑟=2𝑠₁+⋯+2𝑠_𝑛, denote the branch set of the 2-fold branched covering 𝑀_𝑛(𝑟_𝑖/𝑠_𝑖;𝑝_𝑖/𝑞_𝑖;ℎ/𝑘), with 𝑟_𝑖=1 for 𝑖=1,…,𝑛, of 𝕊³ which corresponds to the involution 𝜌 shown in Figure 4(b) (recall that 𝑠_𝑖≥1 for 𝑖=1,…,𝑛). Let 𝐦_𝑖=𝑦_𝑖 be the meridians of the components 𝐾_𝑖 of ℒ^′_𝑛+1 and 𝐦₀=𝑥₀ the meridian of the component 𝐿^′₀ of ℒ^′_𝑛+1. The pair (𝐦_𝑖,𝓵_𝑖), where 𝓵_𝑖 is the longitude of 𝐾_𝑖, is a preferred frame; that is, 𝓵_𝑖∼0 in the exterior space 𝕊³⧵𝐾_𝑖 and 𝑙𝑘(𝐦_𝑖,𝓵_𝑖)=1 for 𝑖=1,…,𝑛. The pair (𝐦₀,𝓵^′₀), where 𝓵^′₀ is the longitude of 𝐿^′₀, is not a preferred frame since 𝓵^′₀∼−(𝑟−1)𝐦₀ in 𝕊³⧵𝐿^′₀, where 𝑟=2𝑠₁+⋯+2𝑠_𝑛. To have a preferred frame, we take the pair (𝐦₀,𝓵₀), where 𝓵₀=𝓵^′₀+(𝑟−1)𝐦₀. Let 𝑉 be a regular neighbourhood of the link ℒ^′_𝑛+1 in 𝕊³. Without loss of generality, we can choose 𝑉, the meridians 𝐦_𝑖, and the longitudes 𝓵_𝑖, 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,…,𝐵_𝑛, say. To obtain the branch set ℒ_𝑟(𝑝_𝑖/𝑞_𝑖;ℎ/𝑘), where 𝑟=2𝑠₁+⋯+2𝑠_𝑛, via the Montesinos algorithm, we isotopy the 𝐵_𝑖’s along the images 𝜋(𝓵_𝑖) of the longitudes 𝓵_𝑖 and replace them by an ℎ/𝑘 rational tangle for 𝑖=0 and by 𝑝_𝑖/𝑞_𝑖 rational tangles, for 𝑖=1,…,𝑛, as in Figure 6.

Figure 5 

The quotient (𝕊3⧵int𝑉)/𝜌.

Figure 6 

The link ℒ𝑟(𝑝𝑖/𝑞𝑖;ℎ/𝑘), 𝑟=2𝑠1+⋯+2𝑠𝑛.

Summarizing, we have proven the following main result.

Theorem 8. Let ℳ=𝑀_𝑛(𝑟_𝑖/𝑠_𝑖;𝑝_𝑖/𝑞_𝑖;ℎ/𝑘), 𝑟_𝑖=1 and 𝑠_𝑖≥1, for 𝑖=1,…,𝑛, be the closed connected orientable 3-manifold obtained by Dehn surgeries on the components of the link ℒ_2𝑛+1. Then ℳ is the 2-fold covering of the 3-sphere branched over the link ℒ_𝑟(𝑝_𝑖/𝑞_𝑖;ℎ/𝑘), where 𝑟=2𝑠₁+⋯+2𝑠_𝑛, pictured in Figure 6.

Theorem 9. Let 𝐾_𝛼/𝛽(𝛾), 𝛾=ℎ/𝑘≠∞, be the closed connected orientable 3-manifold obtained by 𝛾 Dehn surgery on the hyperbolic 2-bridge knot 𝐾_𝛼/𝛽, where 𝛼/𝛽=[−2𝑠₁,2𝑞₁,…,−2𝑠_𝑛,2𝑞_𝑛]. Then 𝐾_𝛼/𝛽(𝛾) is the 2-fold covering of the 3-sphere branched over the link ℒ_𝑟(𝑝_𝑖/𝑞_𝑖;ℎ/𝑘), where 𝑟=2𝑠₁+⋯+2𝑠_𝑛 and 𝑝_𝑖=1 for every 𝑖=1,…,𝑛.

For example, 𝐾_𝛼/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 link ℒ10(1/3,1/4;ℎ/𝑘).

Acknowledgments

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.