MX4540: KNOTS (2018-2019)

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

Course Aims

To show that simple tangible objects, knots and links, can be investigated and classified by mathematical methods, ranging from elementary to reasonably sophisticated.

To develop mathematical tools to express heuristic notions of equality and difference in knots.

To appreciate the connection between logical deduction and computational methods in developing elementary invariants.

To appreciate the need for and use of more sophisticated methods involving elementary topology and groups and hence the interplay between various branches of mathematics.

Learning Objectives

By the end of the course the student should:

understand the idea of invariants of knots; be able to distinguish given knots using the idea of colouring be able to compute and discuss simple invariants such as crossing number, unknotting number, determinant;  be able to use Reidemeister moves to prove the invariance of these and other invariants including polynomial invariants; be able to distinguish given knots by calculating their polynomial or other invariants; appreciate the need for more sophisticated techniques from low dimensional topology

This course alternates with MX4549 Geometry.

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

More Information about Week Numbers

1st Attempt: 1 two-hour written examination (80%) and continuous assessment (20%)

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact the Course Coordinator for feedback on the final examination. Students do practice questions in tutorials allowing formative assessment and feedback from the tutor.

In-course assessment will be marked and feedback provided to the students.