A Knot Invariant is an invariant (e.g., a number, polynomial, Laurent series, ...) that is used to distinguish knots and extract information about their properties. There are several such invariants. Some elementary ones that you may have seen include: tricolorability, chirality and crossing number. More advanced examples include: Alexander Polynomial, Jones polynomial, Kaufman Polynomial, Conway Polynomial, HOMFLY Polynomial, or various other Quantum Knot Invariants.
A large class of knot invariants, of which many of the above are examples, are called Finite Type Invariants (or Vassiliev Invariants). In this project we touch upon these invariants and investigate their properties. Possible avenues to pursue include (in increasing order of difficulty):
1- Computing examples of (lower degree) Vassiliev Invariants and investigating their relation to other more elementary invaraints
2- Construction of Vassiliev Invariants using Weight Systems; Weight Systems associated to Lie algebras
3- Relation to Hopf algebra of Chord Diagrams; Fundamental Theorem (Kontsevitch-Vassiliev)
4- Kontsevitch Integral as a universal Vassiliev Invariant
5- Relation to Topological Quantum Field Theories and Chern-Simons Theory
6- Relation to Braided Monoidal Categories and the Drinfeld associator; geometry of Hyperplane Arrangements, Knizhnik–Zamolodchikov equations; rationality of the Kontsevitch Integral.
Difficulty of the projects:
1: Easy, 2: Moderate, 3&4: Hard, 5&6: Very Hard.
Dedication and hard work will be required (especially for 3–6).
