(in Polish) Kwantowe niezmienniki węzłów

biology
chemistry
mathematics
physics

elective monographs

The prerequisites for the lecture are minimal, as basic algebra and topology courses. All concepts will remain explained during the lecture if necessary.

Remote learning

The course aims to introduce into the theory of quantized and

categorified invariants of knots. After explaining the classical Alexander,

Conway, Jones and HOMFLY-PT polynomials, we will quantize them

with the use of representations of quantum groups and the Reshetikhin-

Turaev functor and categorify to the Khovanov-Lee homology. All technicalities will be explained on the spot when needed. The course is based on selected fragments of the literature listed below. The suggestions for further reading will be provided.

1. Reidemeister theorem

2. Tricolorability, Knot Group

3. Kauffman bracket, Jones polynomial

4. Braid group, Burau representation

5. Temperley-Lieb algebra

6. Jones polynomial through braid representations

7. Khovanov homology, categorification of the Jones polynomial

8. Frobenius algebras and Topological Quantum Field Theories

9. Tangles and Hopf algebras, graphical calculus

10. Quasitriangular, modular, and ribbon Hopf algebras

11. Quantum groups and representations of tangles

12. Coloring ribbon graphs by representations and modular categories

13. Reshetikhin-Turaev invariants of ribbon graphs from quantum groups

14. Knots and 3-dimensional manifolds

15. Physics, chemistry and biology of knots

M. F. Atiyah, The geometry and physics of knots, Cambridge University

Press, Cambridge, 1990.

D. Bar-Natan. On Khovanov’s categorification of the Jones polynomial.

Algebr. Geom. Topol., 2:337–370, 2002.

P. Etingof, O. Schiffmann: Lectures on Quantum Groups. International

Press (2002)

V. Jones, A polynomial invariant for knots via von Neumann algebras,

Bull. Amer. Math. Soc. (N.S.) 12 (1985) 103--111.

L. Kauffman, Knots and physics, World Scientific Publishing, 3rd edition,

1993.

T. Ohtsuki: Quantum Invariants. World Scientific (2001)

V. V. Prasolov and A. B. Sossinsky, Knots, links, braids and 3-Manifolds.

Translations of Mathematical Monographs 154, Amer. Math. Soc.,

Providence, RI, 1997.

N. Reshetikhin and V. Turaev, Ribbon graphs and their invariants

derived from quantum groups, Comm. Math. Phys. 127 (1990) 1--26.

1. Knowledge of the basic concepts of knot theory, including topological, algebraic, and categorial ones.

2. Understanding the basic results of knot theory connecting their invariants with constructions in representation theory, algebra, and noncommutative geometry. In particular, understanding the equivalence of Jones polynomial definitions, derived from seemingly independent points of view.

3. Knowledge of the relationship of knot theory with other scientific disciplines, such as mathematical physics, chemistry, and biology of proteins and DNA, etc.

4. Readiness of the listener for independent reading of contemporary scientific literature in the field.

Active participation in classes.