MATH39022 - 2007/2008
- Title: Knot Theory
- Unit code: MATH39022
- Credits: 10
- Prerequisites: Skills in drawing clear and accurate diagrams, and in performing algebraic manipulations accurately, are a distinct advantage. Sufficient background is provided by the first year course units.
- Co-requisite units: None
- School responsible: Mathematics
- Members of staff responsible:
The aim of the course is to develop geometrical intuition and calculating skills through an attractive and currently active area of pure mathematics. This subject should appeal to anyone with curiosity about how mathematics can help us to understand and describe the three dimensional physical world.
Brief Description of the unit
A mathematical knot is what you get by tying an ordinary knot in a piece of rope and then splicing the ends together. The length and thickness of the rope being unimportant, it is regarded just as a circle embedded in three dimensional space. The simplest knots include the trefoil and figure-eight knots.
This course unit is an introduction to some recent and exciting developments in knot theory. This is a branch of three-dimensional topology, but no knowledge of topology will be needed. We shall be making calculations based on diagrams of knots. The central theme is ‘polynomial invariants of knots and links’. We shall study the classical (1928) Alexander polynomial, and the Jones polynomial discovered by the Fields medallist Vaughan Jones in 1984. From our calculations, we shall be able to draw some conclusions as to whether two diagrams represent the ‘same’ knot or ‘different’ knots.
On successful completion of this course unit students will be able to
- represent knots and links by means of diagrams;
- use their diagrams to calculate invariants, in particular the Conway and Jones polynomials;
- understand the use of Reidemeister moves in invariance proofs;
- handle concepts from three dimensional geometry such as mirror images, braids etc.
Future topics requiring this course unit
Recommended as background for courses in topology or geometry. Some of the material has connections with combinatorics (graph theory) and with analysis (C*-algebras).
- Knots and links in R3, equivalence of knots. Knot diagrams in R2, equivalence of diagrams. Problems of knot theory. Knot invariants. Mirror images. Composition of knots, prime knots, knot tables. [4 lectures]
- The three Reidemeister moves. Colouring knots: the three-colouring property as an example of an invariant. Orientation, signs of crossings. Linking numbers. Writhing, twisting and linking. 
- The Conway polynomial of an oriented link. Recursive calculation from the axioms. Geometric interpretation of coefficients, relation with linking numbers. Invariance of the Conway polynomial. Application to mirror images of links. 
- Braids, equivalence of braids. Representing knots and links by closed braids. Braid groups. 
- Applications of knot theory to biology. The geometry and topology of DNA molecules. Studying enzyme actions using knot theory 
- The Jones polynomial of an oriented link. Recursive calculation from the axioms. This invariant can distinguish mirror images of knots. States model for the Jones polynomial. The Kauffman bracket. Invariance of the Jones polynomial. 
The course does not follow any particular book, but the books below can be recommended as relevant related texts which cover most of the material in the course and much else besides. There are many other books on knot theory, but most of them assume some background in topology, and so they are at a higher level than this course unit.
- Colin C. Adams, The Knot Book, Freeman 1994. This popular book can be approached before taking the course to give a good impression of the subject, assuming no specialised mathematical background. It includes some interesting chapters on applications of knot theory to other sciences.
- Charles Livingston, Knot Theory, Cambridge University Press 1996. This is a very clearly written monograph, which is particularly good on discussion of the relationships between different knot invariants.
- Stephen C. Carlson, Topology of Surfaces and Manifolds, Wiley 2001. This is a well written book which covers most of the material in both MATH31051 and MATH39022 at an elementary level. It is particularly recommended for students taking both course units.
Teaching and learning methods
Two lectures and an examples class each week. In addition students should expect to spend at least four hours each week on private study for this course unit.