The main emphasis in the course is on classical (threedimensional)
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 56 items have been studied, the lecturer and the students will choose,
time permitting, 23 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:
 Knots, links, knot and link diagrams, Reidemeister moves. Basic problems of
knot theory and some general results.
 The AlexanderConway polynomial (axiomatic theory).
 The Kauffman bracket and the Jones polynomial.
 Braids, the braid group as an algebraic and a geometric object. Artin's theorem.
 Links and knots as the closure 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 ThomArnoldVassiliev 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.
 TemperlyLieb algebra and JonesWitten invariants for knots in 3manifolds,
sketch of TQFT (topological quantum field theory) and Feinman path integrals.
 Knots diagrams on squarelined paper and Dynnikov's unknotting algorithm.
 The YangBaxter 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 ZinnJustin).
 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.
