MM574: Knots and their invariants

The Study Board for Science

Teaching language: Danish or English depending on the teacher, but English if international students are enrolled
EKA: N300064102
Assessment: Second examiner: Internal
Grading: 7-point grading scale
Offered in: Odense
Offered in: Spring
Level: Bachelor

STADS ID (UVA): N300064101
ECTS value: 5

Date of Approval: 11-10-2023


Duration: 1 semester

Version: Approved - active

Comment

The course is co-read with MM867.

Entry requirements

The course cannot be chosen if you have passed, registered, or have followed MM867, or if MM867 is a constituent part of your Curriculum.

Academic preconditions

Students taking the course are expected to have knowledge of the material from MM512 (Curves and Surfaces), MM549 (Topology and Complex Analysis) and MM567 (Ring Theory).

Course introduction

Informally, a mathematical knot is the same as a knot on a string with the ends tied together. A knot is mathematically defined as the image of an injective mapping from the circle into three-dimensional space. 

Knot theory plays a central role in modern mathematics, especially in low-dimensional topology, and in modern physics in string theory and quantum field theory. Furthermore, knot theory plays an important role in the theory of topological quantum computing.

The aim of the course is to give an introduction to mathematical knot theory, including an introduction to central knot invariants. Classically, one of the main goals of knot theory is to obtain a classification of knots, that is, to obtain a complete list (without repetitions) of all equivalence classes of knots. Knot invariants play a role in the classification of knots as well as in relation to modern physics and most recently in the development of quantum computing.

In relation to the competence profile of the programme, the course has an explicit focus:

  • To have a basic understanding of the theory of topological spaces and the classification problem for knots.
  • To have a basic understanding of the interaction between algebra and topology as it relates to knot theory.
  • Be able to explain the definition of central knot invariants, including the Jones polynomial.
  • Be able to calculate knot invariants in examples.

Expected learning outcome

To achieve the course objectives, the learning objective of the course is that the student demonstrates the ability to:

  • In a written test, apply terms and concepts from the course syllabus to concrete mathematical examples.
  • Formulate the written presentation in correct mathematical language and argue for the correctness of the results in a mathematically rigorous way

Content

The course contains a selection of the following main areas: 

  • The definition of nodes, links and link diagrams
  • Reidemeister's theorem
  • Classical invariants: unknotting number, crossing number, genus and linking number
  • The classification problem for knots
  • The knot group and Wirtinger's presentation of this
  • The Jones polynomial

Literature

See itslearning for syllabus lists and additional literature references.

Examination regulations

Exam element a)

Timing

Spring and June

Tests

Protfolio exam

EKA

N300064102

Assessment

Second examiner: Internal

Grading

7-point grading scale

Identification

Full name and SDU username

Language

Normally, the same as teaching language

Duration

Take home assignment - 72 hours

Examination aids

Allowed

ECTS value

5

Additional information

Portfolio exam consisting of:
- Mandatory assignment during the course
- Written take home project in June
Both elements have to be passed, and a collective grade is given

With 9 or fewer students the reexamination is changed to an oral exam. No later than 14 days before the exam, a list of possible exams questions will be released, for the students to prepare. At the reexam the student will draw and exam question at random, and will have 30 minutes to prepare. The examiners can ask the student questions regarding the drawn topic, or other topics from the course. The examination will last 30 minutes.

Indicative number of lessons

28 hours per semester

Teaching Method

At the faculty of science, teaching is organized after the three-phase model ie. intro, training and study phase.
  • Intro phase: 28 hours
  • Training phase: 14 hours
Activities during the study phase:
  • Studying the course material and preparing the weekly exercises, individually or through group work
  • Contribute to online learning activities related to the course

Teacher responsible

Name E-mail Department
Jørgen Ellegaard Andersen jea@sdu.dk Quantum Mathematics

Additional teachers

Name E-mail Department City
Du Pei dpei@imada.sdu.dk Institut for Matematik og Datalogi
Jane Jamshidi jaja@sdu.dk Quantum Mathematics

Timetable

Administrative Unit

Institut for Matematik og Datalogi (matematik)

Team at Registration

NAT

Offered in

Odense

Recommended course of study

Transition rules

Transitional arrangements describe how a course replaces another course when changes are made to the course of study. 
If a transitional arrangement has been made for a course, it will be stated in the list. 
See transitional arrangements for all courses at the Faculty of Science.