On Isotopies and hyperbolicity of weaves

TL;DR

This paper characterizes isotopies and hyperbolicity of weaves modeled as links in a thickened torus, providing criteria from diagrams and proving the absence of essential Conway spheres.

Contribution

It introduces a diagram-based characterization of weave isotopies and hyperbolicity, and establishes the non-existence of essential Conway spheres for weaves.

Findings

Weaves can be characterized by their diagrams in the thickened torus.

Hyperbolicity of weaves can be determined from their diagrams.

No essential Conway sphere exists for a weave.

Abstract

A weave is a type of textile that consists of vertical and horizontal threads, and typically it has a periodic structure. In this paper, we regard a weave as a link in the thickened torus with a diagram consisting of closed geodesics. As main results, we characterize isotopies and hyperbolicity of weaves to determine them from diagrams. Moreover, we show that there does not exist an essential Conway sphere for a weave. We use normal positions of essential surfaces of weave complements to describe them.