Program
Seminar on Knot Theory 北京纽结理论研讨班
Student No.:50
Time:Every Monday 13:30-15:00
Instructor:Manturov Vassily Olegovich,Hang Wang  
Place:Conference Room 3, Floor 2, Jin Chun Yuan West Building
Starting Date:2012-9-10
Ending Date:2012-10-29

 

 

 

Date: September 10, 2012

 

 

Instructor: Manturov Vassily Olegovich [People's Friendship University of Russia]

 

Time: Monday 13:30-15:00pm

 

Abstract:

 

During this seminar we will discuss the following problems of knot theory based the first two lectures. The first four are motivating problems for learning knot theory and the last five are exercises.

 

1. Find a graph of a trivial knot diagram, where we cannot apply IB, IIB and III Reidemeister moves to reduce it into a circle (which means that we have to increase the crossing numbers).

 

2. Prove that the three Reidemeister moves are independent of each other (i.e., any one of the moves cannot be obtained from the other two).

 

3. In the equivalence class of each knot, we can find a knot diagram in the plane which has only one crossing.

 

4. For every nontrivial knot in R^3, we can find a line so that the number of intersections of the line with the knot are larger or equal than four.

 

5. Use Reidemeister moves to prove that figure 8 knot is isotopic to its mirror image.

 

6. Use Reidemeister moves to prove that the trefoils with two different orientations are isotopic.

 

7. Calculate the Kauffman bracket of the trefoil knot and prove that it is not isotopic to its mirror image.

 

8. Using color invariant to prove the two diagrams are not isotopic.

 

9. Calculate the linking number of the Whitehead link. Is the Whitehead link trivial or not?

 

 
 
2012年9月10日

 

 

报告人:Manturov Vassily Olegovich [People's Friendship University of Russia]

 

时间:周一下午13:30-15:00

 

摘要:

 

1.请你们找一个平凡纽结的图,使得我们把它变成一个圈的时候,不能用 IB, IIB 和III 移动,(就是说,我们必须增加交叉的个数)。

 

 2.证明三个Reidemeister移动是独立的(就是说任何一个移动不可以用其他两个移动得到)。

 

 3. 在每个纽结的等价类中,可以找到一张纽结的图在平面上的投影只有一个交叉。

 

 4. 对于每一个在R^3中的一个非平凡的纽结的图K,我们都可以找到一条直线,使得这个直线l和图K的交点个数大于等于4.

 

 5. 利用Reidemeister移动说明8纽结和它的镜像是同痕的。

 

 6.利用Reidemeister移动说明具有不同定向的三页纽结是同痕的。

 

 7.计算三页纽结的Kauffman括号,并证明三页纽结和它的镜像是不同痕的。

 

 8. 用颜色不变量证明?和?不同痕。

 

 9. 计算Whitehead链的环绕数(或者:链环数 linking number),请问Whitehead链是不是平凡的?

 

 

 

 

 

 

 

Date: September 17, 2012

 

 

Title: A polynomial invariant of virtual knots.

 

Speaker: Zhiyun Cheng 程志云

 

Institute: School of Mathematical Sciences, Beijing Normal University

 

Abstract:  

 
In this talk I will give a short review of virtual knot theory. A new polynomial invariant, say the odd writhe polynomial will be defined, and some interesting properties of it will be discussed.

 

 

Date: September 24, 2012

 

 

Title: Knot theory and braid. 

 

Speaker: Hang Wang 王航

 
Time: Monday 13:30-15:00

 

Place: Second floor conference

 
 

Abstract:

 
We are going to introduce the theory of braids. Braid and its representation gives us another point of view to knot theory and providing new methods of constructing knot invariants. In the first part of this introductory talk, we are going discuss braid groups and their properties, Markov theorem and Yang-Baxter equation. Then we discuss Markov theory for virtual knots in the second half.

 

 

Date: September 27, 2012.

 

 
Title: Graph-valued invariant polynomials for virtual knots and free knots

(Partially joint work with Louis Hirsch Kauffman)

 

Speaker: Professor Manturov Vassily Olegovich [Peoples' Friendship University of Russia]

 

Organizer: Manturov Vassily Olegovich, Hang Wang.

 

Time: Thursday 13:30-15:00

 

Place: Floor 2 seminar room 3.

 

 
 

Abstract:

 

Our talk will be devoted to generalizations of several quantum link invariants.

We use the pictures of Kuperberg's calculus where invariance under Reidemeister moves follows from some relations of graphs.

 

When graphs are planar, these relations allow one to evaluate these invariants as polynomials, however, when they are not, then in some situation it allows to evaluate (non-ellipltic) graphs as themselves.

 

Thus, if a virtual knot (free knot) diagram is complicated enough then it can serve as a "pattern" which "lives inside" every other diagram of the same knot.

 

This point of view allows one to reduce many problems about virtual knots to problems about their representatives: minimality (and, sometimes recognition), non-commutativity, estimatesfor crossing point numbers etc.

 
We shall address three different graphical calculi corresponding to three Lie algebras.
 

 

 

 
 

Date: October 15, 2012.

 

 

Title: An introduction to knot floer homology

 

 

Speaker: Yinghua Ai 艾颖华 (Tsinghua University) 

 

 

Organizer: Manturov Vassily Olegovich, Hang Wang

 

 

Time: Monday 13:30-15:00.

 

 

Place: Conference Room 1, Floor 1, Jin Chun Yuan West Building

 

 
 

Abstract:

 

 
Knot floer homology was introduced by Ozsvath, Szabo and Rasmussen in 2003, it gives a categorification of the Alexander polynomial. We will give a brief introduction to its definition and survey some of its applications.

 

 
 

Date: October 22, 2012.

 

 

Title: An introduction to Khovanov homology and Milnor conjecture

 

 

Speaker: Professor Jiajun Wang [Peking University]

 

 

Time: Monday 13:00-14:30

 

 

Place: Floor 2 seminar room 3, Jinchunyuan West Building.

 

 
(Please notice that the starting time is 30 min earlier than the usual time for this seminar.)

 

 

 

 

 

Date: October 29, 2012.

 

 

Title: Knot/Link Theory and Local Mirror Symmetry

 

 

Speaker: Professor Jian Zhou 周坚 [Tsinghua University]

 

 

Time: Monday, 1:30-3:00pm

 

 

Place: Conference room 3, Floor 2, Jin Chun Yuan west building

 

 

Abstract:

 

 
In a series of two talks we will make an informal introduction to the spectacular connection between knot/link theory and local mirror symmetry of toric Calabi-Yau 3-folds.
 

 

 

 

Date: November 12, 2012.

 

 

Title: Knot/Link Theory and Local Mirror Symmetry (II)

 

 

Speaker: Professor Jian Zhou 周坚 [Tsinghua University]

 

 

Time: Monday, 1:30-3:00pm

 

 

Place: Conference room 3, Floor 2, Jin Chun Yuan west building

 

 

Abstract:

 

 
In a series of two talks we will make an informal introduction to the spectacular connection between knot/link theory and local mirror symmetry of toric Calabi-Yau 3-folds.