Description

Knots can be studied from a number of points of view. The first method of attack is to pull out the powerful invariant from Topology III: the fundamental group. From this group, one can derive the Alexander polynomial, which has the advantage over a group of being very explicit. Then there are the "quantum" knot polynomials like the Jones polynomial, which at first sight have nothing to do with the fundamental group. Further afield, there is Floer homology or the geometry of the knot exterior. In this project you will explore one or more of these approaches to understanding knots, the interconnectness of the approaches, and some things that can be proved using them.

There is much scope for choosing your own direction with this project - from revisiting the original work of the 1920s by Alexander and others to exploring fashionable knot theoretic ideas from the 21st century.

Prerequisites

Co-requisites

Resources

• "Introduction to knot theory" by Crowell and Fox.
• "The knot book" by Colin Adams.