**Introduction**

The course is devoted to the modern knot theory.

The course starts with basic notions of Reidemeister moves, and proceeds with simplest invariants like colouring invariant, linking numbers, Kauffman bracket form of the Jones polynomial, Conway polynomial, which makes it accessible to students with no prerequisites.

Then it proceeds with basic invariants of knots such as fundamental group and the knot quandle, discusses braid theory (with Alexander and Markov theorem).

The course encompasses such deep constructions as Kontsevich integral and Khovanov homology (including Rasmussen invariant) and ends with author's personal results in this area.

We shall provide many problems: from exercises to unsolved problems in low-dimensional topology.

**Syllabus**

1. Reidemeister moves, colouring invariants, linking number

2. The Kauffman bracket and the Jones polynomial

3. Fundamental group. The knot group

4. The knot quandle. The complete knot invariant

5. The braid group and the braid recognition algoritm

6. Alexander's theorem and Markov's theorem

7. The Alexander polynomial

8. Quadrisecants of knots

9. Vassiliev's invariants. The chord diagram algebra

10. The Kontsevich integral

11. The Khovanov homology

12. The Rasmussen invariant. Sliceness obstructions

13. Introduction to virtual knot theory

14. The Khovanov homology for virtual knots with arbitrary coefficients

15. Free knots and the parity bracket

16. A survey of unsolved problems

**Lecturer Intro**

Vassily Olegovich Manturov, Professor of Moscow Institute of Physics and Technology

**Education:**

• 2008, Habilitation Thesis "Geometry and Combinatorics of Virtual Knots”, M.V. Lomonosov Moscow State University

• 2002, Ph.D. "Bracket Structures in Knot Theory", M.V. Lomonosov Moscow State University

1995-2000, Student, Department of Mechanics and Mathematics, M.V. Lomonosov Moscow State University, Graduated with Excellence in Mathematics.

**Positions:**

• Professor of RAS (elected in 2016),

• Managing Editor of the "Journal of Knot Theory and Its Ramifications”, since 2016,

• Bauman Moscow State Technical University, Full Professor, since November 2010,

• Editor-in-Chief's Deputy for "Proceedings of the Seminar on Vector analysis with its applications to geometry, mechanics, and physics", Moscow State University, in Russian (Proceedings are published since 1930s),

• Member of the Editorial Board of “ISRN Geometry”,

• Member of the Laboratory “Quantum Topology”, Chelyabinsk State University, Chelyabinsk, Russia,

• Member of the Moscow Mathematical Society, Member of the American Mathematical Society,

• Member of the dissertation council of the Kazan State University since 2019.