Reducing Dehn Fillings and Small Surfaces

Abstract

Let M be a compact, connected, orientable 3-manifold with a torus boundary component ∂ 0 M . Let γ be a slope on ∂ 0 M , that is, the isotopy class of an essential simple closed curve on ∂ 0 M . The 3-manifold obtained from M by γ-Dehn filling is defined to be M (γ) = M ∪ V γ , where V γ is a solid torus glued to M along ∂ 0 M in such a way that γ bounds a meridian disk in V γ .By a small surface we mean one with non-negative Euler characteristic including non-orientable surfaces. Such surfaces play a special role in the theory of 3dimensional manifolds. We shall say that a 3-manifold M is hyperbolic if M with its boundary tori removed has a complete hyperbolic structure of finite volume with totally geodesic boundary. Thurston's geometrization theorem for Haken manifolds [22] asserts that a hyperbolic 3-manifold M with non-empty boundary contains no essential small surfaces. Furthermore, if M is hyperbolic, then the Dehn filling M (γ) is also hyperbolic for all but finitely many slopes [22], and a good deal of attention has been directed towards obtaining a more precise quantification of this statement.Let us say that a 3-manifold is of type S, D, A or T if it contains an essential orientable small surface which is an essential sphere, disk, annulus or torus, and of type P, B or K if it contains a non-orientable small surface which is a projective plane, Möbius band or Klein bottle, respectively. In particular, 3-manifolds of type S, D, A and T are called reducible, ∂-reducible, annular and toroidal, respectively. The distance ∆(γ 1 , γ 2 ) between two slopes on a torus is their minimal geometric intersection number. The bound ∆(X 1 , X 2 ) is the least non-negative number m such that if M is a hyperbolic manifold which admits two Dehn fillings M (γ 1 ) and M (γ 2 ) of types X 1 and X 2 , respectively, then ∆(γ 1 , γ 2 ) m. Surveys of the known bounds for various choices (X 1 , X 2 ) and the maximal values realized by known examples are given in [5,7,25].In this paper we consider the case when M (γ 1 ) is of type S or P, and M (γ 2 ) is of type S, P, A, B, T or K, and recover the known bounds for ∆(γ 1 , γ 2 ) in all eleven cases by a unified argument. Suppose that M (γ i ) contains such a small surface F i . Then we may assume that F i meets the attached solid torus V γi in a finite collection of meridian disks, and is chosen so that the number of disks n i is minimal among all such surfaces in M (γ i ). The new results are the restrictions on n 2 in the cases (S/P, A) and (S/P, B) when ∆(γ 1 , γ 2 ) = 2 and in the cases (S, T ) and (S, K) when ∆(γ 1 , γ 2 ) = 3. The main results of this paper are the following.sangyop lee, seungsang oh and masakazu teragaito Theorem 1.1. Suppose that M is hyperbolic. If M (γ 1 ) and M (γ 2 ) are of type S or P, then ∆(γ 1 , γ 2 ) 1.The original proof of the case (S, S) is very complicated [10]. Our proof is remarkably short, although, like [10], it is based on the analysis of intersections of two surfaces. Recently, Hoffman and Matignon [15] also gave a short proof, along almo...

Boundary structure of hyperbolic 3-manifolds admitting toroidal fillings at large distance

“…So, T α separates M(r α ) into two sides X α and Y α , where all Scharlemann cycle faces of G β are assumed to be contained in X α . Proof This is proved on page 214 of [13] in the Proof of proposition 4.1. Proof Assume for contradiction that all Scharlemann cycles in D have the same pair of labels, say, { p, p+1}.…”

Spinal Knots in Lens Spaces

“…The problem is completely solved for torus knots [22] and satellite knots [2,14,24,25]. It is also known that there are many hyperbolic knots which produce lens spaces; among these the (−2, 3, 7)-pretzel knot [6] produces L (18,5) and L (19,7). The following result answers [4, Conjecture C].…”

Reducing and annular Dehn fillings

“…As usual, the assumption that M (π) and M (γ) each contains an essential sphere and an essential annulus with ∆(π, γ) = 2 leads to two labelled graphs in the 2-sphere and the annulus. A result of [19] allows us to assume that the graph in the annulus has exactly two vertices. In Section 5 we collect general lemmas about the two labelled graphs.…”