Combinatorial Knot Theory and the Jones Polynomial

TL;DR

This paper reflects on Vaughan Jones's influential work in knot theory, focusing on the Jones polynomial, its origins, and its broad impact across mathematics and physics, highlighting its role as a unifying invariant.

Contribution

It provides a comprehensive exposition of the development and significance of the Jones polynomial and its generalizations, illustrating its interdisciplinary connections.

Findings

Jones polynomial originated from bracket polynomial model

Connections between Jones polynomial and various mathematical fields

Impact of Jones polynomial on physics and natural sciences

Abstract

This paper is a memory of the work and influence of Vaughan Jones. It is an exposition of the remarkable breakthroughs in knot theory and low dimensional topology that were catalyzed by his work. The paper recalls the inception of the Jones polynomial and the author's discovery of the bracket polynomial model for the Jones polynomial. We then describe some of the developments in knot theory that were inspired by the Jones polynomial and involve variations and generalizations of this invariant. The paper is written in the form of a personal odyssey and with the intent to show different mathematical themes that arise in relation to the Jones polynomial. This invariant can be interpreted in relation to combinatorial topology, statistical mechanics, Lie algebras, Hopf algebras, quantum field theory, category theory and more. In each case the Jones invariant appears as a key example for patterns and connections of these mathematical and physical contexts. It is remarkable to what extent Vaughan Jones' discovery of his polynomial has touched so much mathematics, physics, natural science and so many of our mathematical lives.