MA3022: Knot Theory

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA3022
External Subject Code 100405
Number of Credits 10
Level L6
Language of Delivery English
Module Leader Dr Mathew Pugh
Semester Autumn Semester
Academic Year 2024/5

Outline Description of Module

A knot is a closed curve in three dimensions. How can we tell if two knots are the same?

How can we tell if they are different? We will answer these questions by developing many different "invariants" of knots. It is a pure mathematics topic, drawing on simple techniques from a variety of places, but with an emphasis on examples, computations and visual reasoning.

This topic will follow a set of lecture notes, which students will need to study for themselves each week. The supervision session will discuss any difficulties with the previous week’s reading, the solutions to the problems set, and a brief introduction to the next week’s reading.

Recommended Module: MA2014 Algebra 1: Groups

On completion of the module a student should be able to

· Compute invariants of knots and links, algebraic and polynomial invariants by topological and combinatorial methods.

· Determine whether two diagrams represent different links.

· Demonstrate an understanding of the underlying mathematical theory behind these invariants.

How the module will be delivered

You will be guided through learning activities appropriate to your module, which may

include:

· Weekly face to face classes (e.g. labs, lectures, exercise classes)

· Electronic resources to support the learning (e.g. videos, exercise sheets, lecture

notes, quizzes)

Students are also expected to undertake at least 50 hours of self-guided study throughout the duration of the module, including preparation of formative assessments.

Skills that will be practised and developed

Enhanced capacity to study and gain an understanding of advanced mathematical topics, by means of independent reading.

How the module will be assessed

Formative assessment is carried out by means of weekly problems, with feedback provided during the weekly classes.

The assessment for the module will be a written assignment and an oral examination. The written assignment will consist of an extended problem for which a short report will be submitted at the end of the semester. For the oral exam the students will be given a selection of problems to prepare in advance, and the oral exam would then discuss a couple of these (one chosen by the student, and one chosen by the examiners during the exam).

Assessment Breakdown

Type % Title Duration(hrs)
Written Assessment 50 Knot Theory Coursework N/A
Oral/Aural Assessment 50 Knot Theory Oral Exam N/A

Syllabus content

· An introduction to knots and links, diagrams and Reidemeister moves, orientations, and the linking number

· The colouring of knots, the colouring matrix and determinant, and the colouring group

· The Alexander polynomial

· The Jones polynomial, Kauffman bracket, writhe, and span of the Jones polynomial


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