Knot theory, from past to present

Abstract

In this paper, which is the first paper from a trio on important developments of low dimensional topology in the past 100 years, we review the history and major developments in knot theory. This historic account includes the initial attempts at formulating some of the main questions about knots in a mathematical language, putting the definitions and arguments in a rigorous mathematical framework, and employing tools from other fields of mathematics to extract interesting and intriguing results in knot theory. The study starts from Tait's work in nineteenth century and reviews the important steps taken before the introduction of gauge theory. In particular, we will review the prime decomposition of knots and various polynomial invariants constructed for knots and links. We finish the paper by discussing some of the important conjectures in knot theory which have surprisingly simple statement. We also review some of the recent developments around the aforementioned conjecture, including some theorems including contributions from the author.

Keywords

• knot theory
• low dimensional topology
• knot invariants
• unknotting number
• knot polynomials