Volumes, Lorenz-like templates, and braids

TL;DR

This paper simplifies the classification of links in the 3-sphere by introducing a new braid description using generalized T-links, and provides volume bounds for related 3-manifolds.

Contribution

It offers a new braid description for links in the 3-sphere via generalized T-links and extends algorithms to construct links in Lorenz-like templates.

Findings

Provides a quadratic upper volume bound in terms of trip number.

Establishes an upper bound for the sum of volumes of hyperbolic 3-manifolds.

Simplifies link classification in the 3-sphere using generalized T-links.

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.