The main emphasis of the course is on classical (three-dimensional) knots and their invariants, as well as the interconnections between knot theory and physics. The topics listed below are much more than can be covered in a semester course, and, after the first 5-6 items have been studied, the lecturer and the students will choose, time permitting, 2-3 other topics they wish to study from items 7-16.
Prerequisites
Analysis I, Topology I, Abstract Algebra I.
Curriculum:
- Knots, links, knot and link diagrams, Reidemeister moves. Basic problems of knot theory and some general results.
- The Alexander–Conway polynomial (axiomatic theory).
- The Kauffman bracket and the Jones polynomial.
- Braids, the braid group as an algebraic and geometric object. Artin’s theorem.
- Links and knots as closures of braids. Alexander and Markov theorems.
- The Hecke algebra and Vaughan Jones’ construction of the Jones polynomial.
- Completely solvable models of statistical physics, partition functions, the Jones polynomial as the partition function of the Potts model.
- The Thom–Arnold–Vassiliev philosophy of finite type invariants.
- Vassiliev knot invariants (axiomatic theory and some computations).
- The Kontsevich integral and a sketch of the proof of the existence of Vassiliev invariants.
- Sketch of the construction of the Vassiliev spectral sequence.
- Temperley-Lieb algebra and Jones–Witten invariants for knots in 3-manifolds, a sketch of TQFT (topological quantum field theory) and Feynman path integrals.
- Knot diagrams on square-lined paper and Dynnikov’s unknotting algorithm.
- The Yang-Baxter equation as a machine for producing knot invariants.
- Matrix models in statistical physics and the computation of the number of alternating knots (following Zuber and Zinn-Justin).
- Virtual knot theory: formal (diagrammatic) approach and geometric interpretation (following Kauffman and Manturov).
Textbooks
- Colin C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, 2004.