Knot Theory

The main emphasis in 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: You should be familiar with the basics of the topology of Euclidean space R^n (continuous map, homeomorphism, compactness, path connectedness) and with the definitions of the basic algebraic structures (group, semigroup, ring, module, algebra), although no serious theorems about them will be used.


Curriculum:

  1. Knots, links, knot and link diagrams, Reidemeister moves. Basic problems of knot theory and some general results.
  2. The Alexander--Conway polynomial (axiomatic theory).
  3. The Kauffman bracket and the Jones polynomial.
  4. Braids, the braid group as an algebraic and a geometric object. Artin's theorem.
  5. Links and knots as the closure of braids. Alexander and Markov theorems.
  6. The Hecke algebra and Vaughan Jones' construction of the Jones polynomial.
  7. Completely solvable models of statistical physics, partition functions, the Jones polynomial as the partition function of the Potts model.
  8. The Thom--Arnold--Vassiliev philosophy of finite type invariants.
  9. Vassiliev knot invariants (axiomatic theory and some computations).
  10. The Kontsevich integral and a sketch of the proof of the existence of Vassiliev invariants.
  11. Sketch of the construction of the Vassiliev spectral sequence.
  12. Temperly-Lieb algebra and Jones--Witten invariants for knots in 3-manifolds, sketch of TQFT (topological quantum field theory) and Feinman path integrals.
  13. Knots diagrams on square-lined paper and Dynnikov's unknotting algorithm.
  14. The Yang-Baxter equation as a machine for producing knot invariants.
  15. Matrix models in statistical physics and the computation of the number of alternating knots (following Zuber and Zinn-Justin).
  16. 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.
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