Mario Eudave
UMALCA Special Mention 2004

In previous work (Trans. Amer. Math. Soc. 306 (1988), no. 2, 773{790) MarioEudave combined under one umbrella a collection of recent advances in knot theory.These include both the primeness of unknotting number one knots, and the solution to the band sum problem. The umbrella is this: In how many ways can one attacha rational tangle toa prime tangle and get a nonprime (i.e. composite, split, ortrivial) link? He showed that there are at most three. Furthermore, at most one gives a split link and at most one gives a trivial link.

The paper [1] provides more detail about a particularly important case, that in which a split link is converted to a composite link. Eudave had earlier shown that this can only be done by a band move. Now, by applying sutured manifold theory in a new and imaginative way, he is able to show that the band must cross some decomposing sphere exactly once. The importance of this result is best seen in an immediate corollary: Any strongly invertible knot satisfaces the cabling conjecture. That is, if K is a strongly invertible knot and surgery on K produces a reducible manifold, then K is a cabled knot and the surgery slope is that of the cabling annulus. This further increases the credibility of the cabling conjecture, which has now been verified also for satellite knots.

W. H. Jaco proved that if a 2-handle is added to an irreducible, boundary reducible3-manifold M along an attaching curve whose complement in the boundary of M is incompressible, then the resulting manifold is still irreducible but becomes boundary-irreducible. That is, it contains neither essential spheres nor essential disks. This lemma has proven to be extremely useful in combinatorial 3-manifold theory. It is natural to wonder whether one can similarly determine when attaching a 2-handle creates a manifold without essential tori and annuli (so in particular the manifold supports a hyperbolic structure). In [2] Eudave presents counterexamples to what would seem to be the natural generalization, and is also able to sort out exactly what is true. For example, under the standard hypotheses, he shows that any annulus or torus created by adding a 2-handle can pass at most twice through the 2-handle. As an application, he is able to characterize all satellite knots which have tunnel number one. (This characterization was found independently by K. Morimoto and M. Sakuma, using diferent methods.)

Let K be a knot in the 3-sphere, and let K(p/q) be the manifold obtained by p/q- Dehn surgery on K. In [3] Eudave considers an important and interesting problem:

Suppose K is not a satellite knot . When can an incompressible torus be created after Dehn surgery? Gordon conjectured that if K is not a satellite knot and K(p/q) contains an incompressible torus, then lql is less or equal to 2. For a nonsatellite strongly invertible knot, Eudave proves the following. If K(p/q) contains an incompressible torus, then lql is less or equal to 2 i.e., a positive solution of the Gordon conjecture for strongly invertible knots. Gordon and Luecke also have proved this for all hyperbolic knots, and thus settled the Gordon conjecture.

In [4] Eudave gives a quite interesting infinite family of strongly invertible hyperbolic knots K such that K(m/2) contains an incompressible torus hitting the glued solid torus twice. These knots also admit two integral surgeries producing Seifert fibered manifolds over a 2-sphere with at most three exceptional fibers.

Furthermore, an infinite subfamily of these knots have two more non-hyperbolic surgeries; one is an integral toroidal surgery and the other is an integral Seifert bering surgery. The construction is based on finding a family of very strange tangles (B, t), with the property of having sums with rational tangles which produce the trivial knot, a doubly composite knot (in fact, it is a sum of two Montesinos tangles) and Montesinos knots or links. The 2-fold branched covering space of the 3-ball B branched along the strings t is the exterior of a desired knot K.

For a slope r on a torus boundary of a 3-manifold M, let M(r) be the result of Dehn filling of M along r and let D(r1,r 2) be the minimal intersection number of the two slopes r1 and r2. In [5] Eudave and Wu consider 3-manifolds M with slopes r1 and r2 such that M is hyperbolic but M(r1), M(r2) are not. The main results are: There are infinitely many hyperbolic manifolds M which admit two nonhyperbolic Dehn fillings M(r1), M(r2) such that M(r1) is toroidal and annular, M(r2) is reducible and boundary-reducible, and D(r1,r2)3D2. There are infinitely many hyperbolic manifolds M which admit two nonhyperbolic Dehn fillings M(r1), M(r2) such that M(r1) is reducible, M(r2) is toroidal, and D(r1,r2)3D3. There are infinitely many hyperbolic manifolds M with two boundary components, each of which admits two reducible Dehn fillings M(r1), M(r2) with D(r1,r2) less or equal to 1.Morimoto and Sakuma constructed examples of tunnel number one knots whose complements contain incompressible, non-boundary parallel, tori . In [6] examples are constructed of tunnel number one knots whose complements contain incompressible surfaces of arbitrarily high genus. This answers afirmatively a question first asked by Gordon and Reid.

For any compact 3-manifold M, W. Haken showed that there is an upper bound (depending on M) on the number of disjoint closed incompressible embedded surfaces in M no two of which are parallel. B. Freedman and M. H. Freedman later showed how to extend this to allow surfaces with boundary, provided we first bound the first Betti numbers of the surfaces involved. In the context of Haken's result, no bound independent of M is possible, even if we place an upper bound on the Heegaard genus of M|the minimal genus of a surface splitting M into a pair of compression bodies [M. Eudave, Bol. Soc. Mat. Mexicana (3) 6 (2000), no. 2, 263{277]. However, if we bound both the Heegaard genus of M and the topological complexity of the surfaces, then in [7] Eudave and Shor show that such a universal bound exists. Specifically, they show that for any g and b, there is a constant C(g,b) such that if M is a compact 3-manifold of Heegaard genus at most g, and contains a collection S of C(g,b) disjoint compact incompressible surfaces, each having first Betti number at most b, then at least two of the surfaces are parallel.

References
[1 ] Band sums of links which yield composite links. The cabling conjecture for strongly invertible knots, Trans. Amer. Math. Soc. 330 (1992), 463-501.
[2 ] On non-simple 3-manifolds and 2-handle addition, Topology and its applications 55 (1994), 131-152.
[3 ] Essential tori obtained by surgery on a knot, Pac. J. Math. 167 (1995), 81-116.
[4 ] Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, Proceedings of the 1993 Georgia International Topology Conference (W. Kazez, ed.), AMS/IP Studies in Advanced Mathematics 2, vol.1 (1997), 35-61. 3
[5 ] (with Ying-Qing Wu) Nonhyperbolic Dehn ¯llings on hyperbolic 3-manifolds, Pac. J. Math. 190 (1999), 261-275.
[6 ] Incompressible surfaces in tunnel number one knot complements, Topology and its applications 98 (1999), 167-189.
[7 ] (with J. Shor) A universal bound for surfaces in 3-manifolds with a given Heegaard genus, Algebraic and Geometric Topology 1 (2001), 31-37.
[8 ] On hyperbolic knots with Seifert ¯bered Dehn surgeries, Topology and its applications 121 (2002), 119-141.
[9 ] (with Max Neumann-Coto) Acylindrical surfaces in 3-manifolds and knot complements, to appear in Bol. Soc. Mat. Mex