Volumes, Lorenz-like templates, and braids

Abstract

In this paper, we find a more straightforward problem that is equivalent to one of the major challenges in knot theory: the classification of links in the 3-sphere. More precisely, we provide a simpler braid description for all links in the 3-sphere in terms of generalised T-links. With this, we translate the problem of classifying links in the 3-sphere into a problem of counting the number of generalised T-links that represent the same link. Generalised T-links are a natural generalisation of twisted torus links and Lorenz links, two families of links that have been extensively studied by many people. Moreover, we generalise the bunch algorithm to construct links embedded in universal Lorenz-like templates and provide an upper volume bound that is quadratic in the trip number. We use the upper bound obtained from the generalised bunch algorithm for generalised T-links to establish an upper bound for the sum of the volumes of the hyperbolic pieces of all closed, orientable, connected 3-manifolds.