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Lectures "Invariants and pictures"
-- V.O. Manturov
This lecture is given bilingually -- in English and in Russian
Lecture: 18:35 -- 20:00 (GMT+3) on Friday
Zoom link: 569 915 0694
Passcode: 901764
Seminar: 18:35 -- 20:00 (GMT+3) on Wednesday
Zoom link: 780 361 3594
Passcode: 141592
Lectures on Knot theory in Chinese
-- V.O. Manturov

Lecture: 15:00 -- 16:30 (GMT+8) on Sunday
Tencent link: 486 4356 1964
Passcode: 179325
Lectures on Knot theory
-- V.O. Manturov, S. Kim (

Lecture: 09:00 -- 10:25 (GMT+3) on Friday
Zoom link: 569 915 0694
Passcode: 901764
Seminar: 10:45 -- 12:10 (GMT+3) on Friday
Zoom link: 780 361 3594
Passcode: 141592


"Теория узлов" и "Инварианты и картинки"

Уважаемые студенты,

Занятие Теория узлов, 4-мерная геометрия и топология, Группа 2021-2022 (Мантуров Василий Олегович) состоится по вторникам 17:05 - 20:00.

Войти в него можно по ссылке:

До встречи!

Knot Theory in Jilin university

Prof. V.O. Manturov will give lectures on knot theory

from 06/01/2021 at 13:00 (GMT+3), 18:00 (GMT+8) (in Chinese).

Lecture 1. Reidemeister moves. Colouring invariants, the linking number

报告时间:2022/01/06 18:00-19:00

Lecture 2. The Kauffman bracket, the Jones polynomial.

报告时间:2022/01/09 18:00-19:00

Lecture 3. Fundamental group. The knot group.

报告时间:2022/01/13 18:00-19:00

Lecture 4. The knot Quandle is a complete knot invariant.

报告时间:2022/01/16 18:00-19:00

Lecture 5. The braid groups

报告时间:2022/01/20 18:00-19:00

Lecture 6. The Alexander polynomial.

报告时间:2022/01/23 18:00-19:00

Lecture 7. Vassiliev invariant

报告时间:2022/01/27 18:00-19:00

Lecture 8. Khovanov homology

报告时间:2022/01/30 18:00-19:00



Meeting ID : 818 3620 2175

Link :


"Knot theory" and "Invariants and pictures"

Dear colleagues,

On Tuesday, online lectures on "Knot theory" and "Invariants and pictures" are given by professor V.O. Manturov.

  • Теория узлов. Дополнительные главы / Knot Theory. Additional Chapters, Tuesday at 17:05 in Moscow

  • Инварианты и картинки / Invariants and Pictures, вторник Tuesday at 18:35 in Moscow



Lectures "Knot theory" in Tsinghua university

Thursday & Sunday 15:20-16:55, May 12-July 7,2022

Speaker:Prof. Vassily Olegovich Manturov (Russian Academy of Sciences)

Venue:Zoom Meeting ID: 276 366 7254 Passcode: YMSC

Course schedule:

Lecture 1. Reidemeister moves, colouring invariants, linking number

Lecture 2. The Kauffman bracket and the Jones polynomial

Lecture 3. Fundamental group. The knot group

Lecture 4. The knot quandle. The complete knot invariant

Lecture 5. The braid group and the braid recognition algoritm

Lecture 6. Alexander's theorem and Markov's theorem

Lecture 7. The Alexander polynomial

Lecture 8. Quadrisecants of knots

Lecture 9. Vassiliev's invariants. The chord diagram algebra

Lecture 10. The Kontsevich integral

Lecture 11. The Khovanov homology

Lecture 12. The Rasmussen invariant. Sliceness obstructions

Lecture 13. Introduction to virtual knot theory

Lecture 14. The Khovanov homology for virtual knots with arbitrary coefficients

Lecture 15. Free knots and the parity bracket

Lecture 16. A survey of unsolved problems

Satellite conference of ICM2022

"Knot theory and Applications"

June 29 – July 4, 2022

 A satellite conference of ICM2022 Knot theory and Applications will be held on  June 29 – July 5, 2022 online as a fully virtual event.



4-dimensional geometry and topology 

in Jilin university

Prof. V.O. Manturov will give

lectures on 4-dimensional geometry and topology

from 08/01/2022 at 14:00 (GMT+3), 19:00 (GMT+8) (in English).

ZOOM ID: 810 4009 0103,PW: 989134


Lecture 1. 2023.01.08

The Poincare conjecture h-cobordism theorem

We formulate and proof the famous h-cobordism theorem by Smale

and the Poincare conjecture in dimensions greater than or equal to 5

Lecture 2. 2023.01.13

Freedman's proof of the Poincare conjecture in dimension 4.

We introduce Casson's handles and prove the topological Poincare

conjecture in dimension 4.


Lecture 3. 2023.01.15

Intersection forms.

We discuss algebraic classification of Z-valued quadratic forms.

We shall consider simply connected 4-manifolds and formulate

some classification results based on the intersection pairings on

second homology groups.

Lecture 4. 2023.01.20

Theorems of Wall and Rokhlin

We prove classical theorems of Wall (about stable equivalence of manifolds) and Rokhlin (about signatures and cobordidsms)


Lecture 5. 2023.01.27

Preliminary materials: spin structures, Clifford algebras

We introduce necessary algebraic background needed for formulation

of Donaldson and Seiberg-Witten invariants


Lecture 6. 2023.01.29

Donaldson's theorem

We sketch the proof of Donaldson's theorem that if an intersection

form of a smooth manifold is positive-definite then it is diagonalisable.


Lecture 7. 2023.02.03

Seiberg-Witten invariants

We give a quick introduction into Seiberg-Witten invariants and list results: short proof of Donaldson's theorem, triviality of SW results for positive curvature, exotic smooth structures


Lecture 8. 2023.02.05

Exotic R^{4}.

We discuss the approaches to constructing smooth structures on R^{4}

due to Freedman, Gompf, Taubes, including the celebrated Taubes'

theorem on continuously many different smooth structures

Time line

Chicago -8h

Moscow +0h

Beijing +5h

Seoul +6h

Tokyo +6h

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