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Introduction to Knot theory and some related topics


Academic Lectures

Reporter: Prof. Andrei Vesnin (Sobolev Institute of Mathematics, Novosibirsk, Russia)

Title: Introduction to Knot theory and some related topics


Abstract: This is an introductory course in Knot Theory. The theory started in the 19th  century with naive tables of knots, i.e. continuous loops in the space. At present knot theory is a wide area of mathematics having deep relations with front lines of topology, geometry, algebra, combinatorics, theoretical physics and DNA. This areas has a specialized mathematical journal - Journal of Knot Theory and Its Ramifications.

In the class we will discuss basic ideas and classical theorems. We will construct various invariants and use them to distinguish knots. We will complete the course with modern ideas and results.


Time: Sep 13, 2017-Oct 25,2017

Location: A#1138 room, Innovation Park Building 


Lecture 1: Sep 13 (Wed) 10:05 - 11:40 AM

Knots, links and their diagrams. Equivalence of knots. Reidemeister theorem.


Lecture 2: Sep 15 (Fri)  8:00 -   9:35 AM

Knot invarinats from diagrams. Colorings of knot diagrams.


Lecture 3: Sep 20 (Wed)  10:05 - 11:40 AM

Kauffman bracket polynomial. Jones polynomial.


Lecture 4: Sep 22 (Fri)       8:00 -  9:35 AM

Skein relations. Properties of Jones polynomial.


Lecture 5: Sep 27 (Wed)  10:05 - 11:40 AM

Span of Jones polynomial Tait conjecture. HOMFLYPT polynomial.


Lecture 6: Oct 11 (Wed)  10:05 - 11:40 AM

Artin braid groups, Linear representations of groups.


Lecture 7: Oct 13 (Fri)  08:00 -  9:35 AM

Goemetrical braids. Alexander theorem. Markov theorem.


Lecture 8: Oct 18 (Wed)   10:05 - 11:40 AM

Representations of braid groups. Automorphisms of free groups.


Lecture 9: Oct 20 (Fri)  8:00 -   9:35 AM

Briad groups and Yang-Baxter equation. R-matrices.


Lecture 10: Oct 25 (Wed)  10:05 - 11:40 AM

Generalizations and new approaches.


The brief introduction to the reporter: Prof. Andrei Vesnin is head of the Laboratory of Applied Analysis, Sobolev Institute of Mathematics, Siberian Branch of Russian Academy of Sciences and a professor of Geometry and Topology, Novosibirsk State University. He received a Candidate of Sciences in physics and mathematics is 1991 from Sobolev Institute of Mathematics for the thesis ”Discrete groups of reflections and three-dimensional manifolds”, and a Doctor of Sciences in physics in mathematics in 2005 for the thesis ”Volumes and isometries of three-dimensional hyperbolic manifolds and orbifolds”. He was a visiting professor in Seoul National University in 2002 – 2004. 

    Prof. Vesnin's reseach interests include low-dimensional topology, knot theory, hyperbolic geometry, combinatorial group theory, graph theory and applications.

    In 2008 Prof. Vesnin was elected to corresponding member of the Russian Academy of Sciences. 

    Prof. Vesnin is the editor-in-chief of Siberian Electronic Mathematics Reports and a member of editorial boards of Siberian Mathematical Journal and Scientiae Mathematicae Japonicae.