(in Polish) Kwantowe niezmienniki węzłów 1000-1M20KNW
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
|
Term 2023L:
1. Reidemeister theorem |
Type of course
Mode
Prerequisites (description)
Course coordinators
Main fields of studies for MISMaP
Term 2023L: chemistry biology mathematics physics | General: mathematics chemistry biology physics |
Learning outcomes
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.
Assessment criteria
Active participation in classes.
Bibliography
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.
|
Term 2023L:
M. F. Atiyah, The geometry and physics of knots, Cambridge University |
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: