Knots are closed strings in three dimensional space. The fundamental question is to decide when two given knots are the same or if a particular knot is equivalent to another or even knotted at all. Knots have been studied by mathematicians for over a century but in the last 25 years a number of new simple ideas have contributed to remarkable breakthroughs which have helped clear up a large number of outstanding problems and conjectures. These ideas have come from a number of branches of mathematics and not only have influenced knot theory itself but have revolutionised several branches of mathematics and even mathematical physics. Applications have also been found in biology in understanding how DNA strands are knotted. This course is an elementary introduction to modern knot theory as it now stands and some of the tools which are now available for understanding knots. The style and emphasis is on using and understanding the tools rather than a traditional definition-theorem-proof approach.

Pre-requisite Module: MA0212 Linear Algebra

• 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.

27 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Knot and link diagrams will be projected on a screen with copies maintained on a web page. 

Students are also expected to undertake at least 50 hours private study including preparation of solutions to given exercises.

Skills:
Problem solving and algebraic and geometric manipulation.

Transferable Skills:
Mathematical formulation of geometric ideas and problems.

Formative assessment is carried out by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures.  

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of three from four equally weighted questions.

• Curves in the plane, knots, links and ribbons in three dimensional space, link diagrams in the plane, Reidemeister moves, isotopy - ambient and regular isotopy.

• Examples of knots and links including Hopf, trefoil, figure 8, Whitehead, toric, satellite, hyperbolic knots.

• Orientation of links, writhe, linking number, colourability of link diagrams and colouring groups.

• Polynomial invariants of links
  • combinatorial approach to the bracket polynomial,
  • Jones polynomial, Kauffman polynomial, Homfly polynomial, Alexander polynomial, Conway polynomial.

• Graphical description of matrix algebras and matrix operations

• Polynomial invariants
  • algebraic approach via R-matrices and the Yang Baxter equation.

• Tangles and braids.

Knots and Physics, Kauffman, L., World Scientific Press

Knots and Surfaces, Gilbert, N. D., & Porter, T., OUP

The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, Adams, C. C., W. H. Freeman & Company

Knots and Links; Cromwell, P., CUP

An Introduction to Knot Theory, Lickorish, W. B. R., Springer

Quantum Symmetries on Operator Algebras, Evans, D. E., & Kawahigashi, Y., OUP