Mario Eudave-Muñoz

On hyperbolic knots with Seifert fibered Dehn surgeries

Abstract J.C. Dean has described a construction which produces knots having a small Seifert fibered manifold Dehn surgery. We translate this construction to tangles, and show that a trivializable tangle satisfies certain conditions if and only if the corresponding covering knot satisfies Deans conditions. This can be used to show that all known examples of hyperbolic knots producing Seifert fibered manifolds, which were constructed by Montesinoss trick, can in fact be obtained by Deans construction. We extend Deans construction slightly, in order to include surgeries producing Seifert fibered manifolds with orbit surface a projective plane and two exceptional fibers, and we show an infinite family of hyperbolic knots having that kind of surgery.

Band Sums of Links which Yield Composite Links. The Cabling Conjecture for Strongly Invertible Knots

We consider composite links obtained by bandings of another link.It is shown that if a banding of a split link yields a composite knot then there is a decomposing sphere crossing the band in one arc, unless there is such a sphere disjoint from the band. We also prove that if a banding of the trivial knot yields a composite knot or link then there is a decomposing sphere crossing the band in one arc. The last theorem implies, via double branche covers, that the only way we can get a reducible manifold by surgery on a strongly invertible knot is when the knot is cabled and the surgery is via the slope of the cabling annulus

Incompressible surfaces in tunnel number one knot complements

Abstract A knot k in S 3 has tunnel number one, if there exist an arc τ embedded in S 3 , with k ∩ τ = ∂τ , such that S 3 −int N(k∪τ) is a genus 2 handlebody. In this paper we construct for each integer g ≥2, infinitely many tunnel number one knots, whose complement contain a closed incompressible surface of genus g .

On nonsimple 3-manifolds and 2-handle addition

Abstract Suppose a 2-handle is attached to an orientable, irreducible 3-manifold M along a curve α contained in ∂ M , obtaining a 3-manifold M α . Suppose ∂ M is compressible, but ∂ M — α is not. It is proved that if M α is nonsimple, i.e., it contains an essential annulus or torus, then either M is also nonsimple or M α contains an annulus or torus which intersects the 2-handle twice. As an application of the main theorems, all satellite knots in S 3 with tunnel number one are found; they are certain satellites of torus knots.

Primeness and Sums of Tangles

We consider knots and links obtained by summing a rational tangle and a prime tangle. For a given prime tangle, we show that there are at most three rational tangles that will induce a composite or splittable link. In fact, we show that there is at most one rational tangle that will give a splittable link. These results extend Scharlemanns work.

Meridional surfaces and (1,1)-knots

We determine all (1,1)-knots which admit an essential meridional surface, namely, we give a construction which produces (1,1)-knots having essential meridional surfaces, and show that if a (1,1)-knot admits an essential meridional surface, then it comes from the given construction.

Haken manifolds obtained by surgery on closed pure 3-braids

Abstract We consider manifolds obtained by Dehn surgery on closed pure 3-braids in S 3 , and show that for a large class of these links, and for most surgeries, we cannot get a homotopy 3-sphere; in fact in many cases we obtain Haken or laminar manifolds.

A universal bound for surfaces in 3-manifolds with a given Heegaard genus

It is shown that for given positive integers g and b ,t here is a numberC(g;b), such that any orientable compact irreducible 3-manifold of Heegaard genus g has at most C(g;b) disjoint, nonparallel incompressible surfaces with rst Betti number b1 <b . AMS Classication 57N10; 57M25