MX4540: KNOTS (2020-2021)

Last modified: 24 Jun 2020 14:31

Course Overview

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? This course answers these questions by developing many different "invariants" of knots. It is a pure mathematics course, drawing on simple techniques from a variety of places, but with an emphasis on examples, computations and visual reasoning.

Course Details

Study Type	Undergraduate	Level	4
Session	Second Sub Session	Credit Points	15 credits (7.5 ECTS credits)
Campus	Aberdeen	Sustained Study	No
Co-ordinators	Dr Richard Hepworth

What courses & programmes must have been taken before this course?

Either Programme Level 3 or Programme Level 4
Either MX3036 Metric and Topological Spaces (Studied) or MX3532 Metric and Topological Spaces (Passed)
Any Undergraduate Programme (Studied)

What other courses must be taken with this course?

None.

What courses cannot be taken with this course?

None.

Are there a limited number of places available?

Course Description

Knots have been studied mathematically since the 19th century, and knot theory connects with many other areas of pure maths and even theoretical physics. This course concentrates on knot invariants: numbers, polynomials or groups that try to "measure" properties of the knots. Classic invariants such as the colouring group will lead to more modern ones like the Jones polynomial, which was only discovered in the mid-1980s.

Topics will include the following:

Knots and links.
Linking number.
Colourings,determinants and the colouring group.
The Alexander polynomial.
The Jones polynomial.
The Genus.

Syllabus

Knots and links
Diagrams and Reidemeister moves
Sums, mirrors and reverses
The linking number
Colourability
The determinant
The colouring group
Alexander Polynomial
Jones polynomial: definition, computations, properties
Jones polynomial and alternating links
The genus of a knot and prime knots

Contact Teaching Time

Information on contact teaching time is available from the course guide.

Teaching Breakdown

More Information about Week Numbers

Details, including assessments, may be subject to change until 31 August 2023 for 1st half-session courses and 22 December 2023 for 2nd half-session courses.

Summative Assessments

Online forum contribution (assessed through out the course) - 25%

Group project (11 weeks) - 25%

2 x course assessments (1 week each) - 25% each

Alternative Resit Arrangements for students taking course in Academic Year 2020/21

For each assessment on which a student scored E1 or less, the student will complete an appropriate resit assessment, as detailed below. The course score will then be determined based on the weighted average of the resit assessments, together with those original assessments that scored D3 or above.

Course Assessments:
The student will complete a resit assessment of the same format and covering the same material, again with 1 week to complete the assessment.

Online forum contribution:
The student will be allowed, over the course of 1 week, to make new contributions to the discussion forum, and to make a new choice of the 5 discussions they will submit to the forum.

Group Project:
The student will be given a tailored homework-style assessment, based on the topic of their group project, with 2 weeks to complete the assessment.

Alternative Resit Arrangements for students who took the course in Academic Year 2019/20 and C8 students