Core curves of triangulated solid tori

Abstract

One of the most powerful tools in 3-manifold topology is normal surface theory. Many topologically relevant surfaces can be placed into normal form (or some variant of this) with respect to any triangulation T of the manifold M . An important case of this phenomenon is when M is the solid torus and the surface is a meridian disc. The existence of a meridian disc in normal form and the fact that this can be algorithmically detected is the key to Haken’s solution to the problem of recognizing the unknot [1]. However, normal surface theory suffers from some substantial limitations, possibly the most important of which is that every normal meridian disc may have ‘exponential complexity’. More precisely, if n is the number of tetrahedra in T , then normal surface theory only produces a meridian disc D with at most 2 normal triangles and squares, where k = 10 (see [2]). This is more than just an artifice of the theory, because one can find triangulations of the solid torus where every normal meridian disc has exponentially many squares and triangles. Indeed, we will do this explicitly in Section 6. However, in this paper, we show that for a related problem, there is a solution with linearly bounded complexity. In addition to the meridian disc, the solid torus contains another important sub-object: the core curve C, which is defined, up to ambient isotopy, to be {∗} × S ⊂ D × S, where ∗ is a point in the interior of D. In this paper, we will address the problem of placing C into ‘normal form’. The surprising conclusion is that this can be achieved with a linear upper bound on the ‘complexity’ of C.