On Hyperbolic 3‐Manifolds Obtained by Dehn Surgery on Links

Kim, Soo Hwan; Kim, Yangkok

doi:10.1155/2010/573403

International Journal of Mathematics and Mathematical Sciences

2010

DOI: 10.1155/2010/573403

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On Hyperbolic 3‐Manifolds Obtained by Dehn Surgery on Links

Soo Hwan Kim¹,

Yangkok Kim

Abstract: We study the algebraic and geometric structures for closed orientable -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 -fold cyclic covering of -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 -mani… Show more

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“…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][6][7][8][9][10]). This paper relates these methods to study a new class of hyperbolic orientable 3-manifolds via combinatorial tools.…”

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Fundamental Group and Covering Properties of Hyperbolic Surgery Manifolds

2013

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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. Manifolds Obtained by Dehn SurgeriesAs 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][6][7][8][9][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 L 2 +1 be the oriented link with 2 + 1 components 0 , , and , = 1, . . . , , in the oriented 3-sphere S 3 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 ≥ 1. Let us consider the closed connected orientable 3-manifolds ( / ; / ; ℎ/ ) obtained by Dehn surgery on S 3 along the oriented link L 2 +1 such that the surgery coefficients / , / , and ℎ/ correspond to the oriented components , , and 0 , respectively, where = 1, . . . , . Of course, we always assume that gcd( , ) = 1, gcd( , ) = 1, and gcd(ℎ, ) = 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 ...

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Fundamental Group and Covering Properties of Hyperbolic Surgery Manifolds

2013

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A generalization of a formula due to Kummer^†

Choi

Rathie²,

Srivástava

2011

Integral Transforms and Special Functions

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On Hyperbolic 3‐Manifolds Obtained by Dehn Surgery on Links

Cited by 2 publications

References 11 publications

Fundamental Group and Covering Properties of Hyperbolic Surgery Manifolds

Fundamental Group and Covering Properties of Hyperbolic Surgery Manifolds

A generalization of a formula due to Kummer^†

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On Hyperbolic 3‐Manifolds Obtained by Dehn Surgery on Links

Cited by 2 publications

References 11 publications

Fundamental Group and Covering Properties of Hyperbolic Surgery Manifolds

Fundamental Group and Covering Properties of Hyperbolic Surgery Manifolds

A generalization of a formula due to Kummer†

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A generalization of a formula due to Kummer^†