Injectively Immersed Tori in Branched Covers over the Figure Eight Knot

The importance of this result lies primarily in the fact [HLM] that the figure eight knot is universal. That is, that all closed, orientable 3-manifolds are obtainable as branched covers over S, branched over the knot. We also obtain results about incompressible (embedded) tori in some cases. Existence or nonexistence of injectively immersed tori is critical to the understanding of a 3-manifold in light of the Jaco-Shalen and Johannson torus decomposition [J-S,Jh] which plays such a pivotal role in Thurston’s geometrization conjecture and the recent result of Gabai and Casson (independently, see [Ga]) which, when combined with earlier results of Mess and Scott (see [M],[Sc]) shows that a 3-manifold which admits an injectively immersed torus, but no incompressible surfaces, must be Seifertfibered. Our approach to developing this algorithm will be partly geometric and partly combinatorial in nature. We will first describe a fixed geometric structure on S which lifts in a particularly nice way to a geometric structure on any branched cover of S, branched over the figure eight knot, which has the required minimum branching index. We will then use this geometric structure to gradually reduce the problem of finding injectively immersed tori in this branched cover to a problem of finding paths in a certain graph which satisfy certain easily verified conditions. We will also obtain an a priori upper bound on the length of paths which must be considered, so that our algorithm can not only find tori when they exist, but also ascertain when they do not. The overall organization of this paper is as follows: the geometric structures that we will need (Euclidean cone manifold structures) are described in Section 1. Section 2 discusses the particular cone manifold structure on S which will be lifted to the branched covers of S. Section 3 describes the first dimensional reduction that we will need, reducing questions about tori in 3-manifolds to questions about geodesics in 2-manifolds. Section 4 describes how to make the actual calculations needed in section 3 from combinatorial data