Introductory description

A knot may be regarded as a continuous loop of (thin rubber) string. There are two fundamental problems: Is the loop really knotted? When is a loop attainable from another by continuous deformation?

The problem is tackled by computing invariants. If for instance we have a computable way to assign invariant numbers to knots then two knots with different numbers cannot be equivalent. Another approach is to look at the topology of the complement of the knot. Can we find a surface with the knot as boundary? What properties does it have?

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Introduction of knots and knot diagrams
Reidemeister moves
Knot colourings and colouring groups
The Alexander Polynomial
Braids and Braid groups
Seifert circles
The Jones Polynomial
The Conway Polynomial

Learning outcomes

By the end of the module, students should be able to:

• Understand the definition of a knot and draw knot diagrams
• Understand the theory about, and be able to calculate, colouring numbers of knots, as well as colouring groups
• Understand the theory about, and be able to calculate, the Alexander, Jones, and Conway Polynomials for knots
• Understand how knots can be represented using bridges, plats, and braids,

Indicative reading list

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, 1994. 306p
Livingston, Charles. Knot Theory Washington, DC: Math. Assoc. Amer., 1993. 240 p.
Cromwell, Peter Knots and Links Cambridge University Press, 2004. 346 p.

View reading list on Talis Aspire

Subject specific skills

Ability to draw knots and apply Reidemeister moves.
Knowledge of, and ability to calculate and apply, knot invariants.
Ability to represent knots using braids, and to obtain Seifert circles.

Transferable skills

Students will develop their ability to work with abstract concepts, and to explore the use of various tools to distinguish objects.