A personal account of the discovery of hyperbolic structures on some knot complements

TL;DR

This paper recounts the historical development of discovering hyperbolic structures on knot complements, highlighting early work on knot group representations and interactions with William Thurston.

Contribution

It provides a personal historical account of the discovery process of hyperbolic structures on knot complements, emphasizing early research and key meetings.

Findings

Early work on knot group representations into matrix groups

Meeting with William Thurston in 1976 influenced the development

Historical perspective on hyperbolic structures in knot theory

Abstract

I give my view of the early history of the discovery of hyperbolic structures on knot complements from my early work on representations of knot groups into matrix groups to my meeting with William Thurston in 1976. (This article was written by Robert Riley about ten years before his death in 2000 and never submitted for publication. An explanation of why it is being published now and some information about Riley and this article is given in the article by Brin, Jones and Singerman which accompanies this article in this issue of the [journal | arxiv].)