Семинар по теории узлов

УЛК МГТУ им. Н. Э. Баумана (Москва, Рубцовская наб., 2/18), г.Москва

Кафедра проводит семинар по теории узлов под руководством профессора В.О.Мантурова.

Основные направления деятельности семинара:

• теория узлов и маломерная топология, в частности
• топологическая квантовая теория поля,
• квантовые инварианты;
• представления групп и алгебр Ли;
• комбинаторика;
• алгебраическая топология;
• теория минимальных сетей

Руководитель: Проф. РАН В.О.Мантуров Этот адрес электронной почты защищён от спам-ботов. У вас должен быть включен JavaScript для просмотра.

Ближайший доклад

6,13,20,27 октября 2017 года в пятницу, 15:30(UTC+3), ауд.1025, УЛК:

Title: Lectures on Delta structures

Wu Jie

National University of Singapore

Профессор Ву Це (Сингапур) прочтёт четыре лекции о гомотопических группах сфер, \Delta -структурах  и теории кос.

Abstract:  The lectures will give an introduction to simplicial groups as well as to the simplicial structures on geometric groups. The simplicial homotopy is highlighted and combinatorial descriptions of homotopy groups of spheres are stated. Connections between homotopy groups and geometric groups are also given.

Прошедшие семинары

Title: Introduction to Knots, Knotoids and Detecting the UnKnot

Louis H. Kauffman, UIC

25 November 2016, 18:00 at BMSTU

Abstract : This talk is joint work with Neslihan Gugumcu.

Knots and links are topological types of embeddings of circles in three dimensional space.

This talk is concerned with a very specific question: Does the Jones polynomial detect the unknot?

This question has been open since the Jones polynomial was discovered in 1983 and it received a particular focus in the form of the bracket state summation

model for the Jones polynomial. We focus on the diagrammatic bracket state summation. The bracket state summation has been generalized

by Mikhail Khovanov to a homology theory for knots and links known as Khovanov homology. By arranging the bracket states of a knot or link diagram $K$ in the form of a category $Cat(K)$ where

the objects of the category are the bracket states and the generating morphisms are arrows between states, Khovanov is able to make a homological measure of this category that is topologically invariant. In a nutshell, he constructs a functor $F$ from $Cat(K)$ to a Frobenius module category

and takes the cohomology of the category $Cat(K)$ with coefficients in the sheaf defined by the functor $F.$

Khovanov homology has been proven to detect the unknot by Kronheimer and

Mrowka. A graded Euler characteristic of the Khovanov homology reproduces the Jones polynomial.

There remains a huge gap between this stellar result of Kronheimer and Mrowka for Khovanov homology detecting the unknot, and the possiblity that the Jones polynomial itself detects the unknot.

Kronheimer and Mrowka succeed via gauge theory and a comparison of Khovanov homology with Instanton Knot Floer Homology.

It remains to be seen whether this comparison can be extended to the subtler comparison with the Jones polynomial itself.\\

We are investigating a generalization of the conjecture that the Jones polynomial detects the unknot.

A {\it knotoid} is an equivalence class of a planar knot diagram with two free ends. The ends are allowed to be in different regions of the diagram. Knotoid diagrams

are taken up to Reidemeister moves.  The moves are not allowed to take an arc across either of the ends of the knotoid.

The bracket polynomial and hence the Jones polynomial can be immediately  extended to knotoids. We conjecture that {\it the Jones polynomial detects the unknotted knotoid.}.

This conjecture has the same level of plausibility as the original conjecture that the Jones polynomial detects the unknot. The knotoid conjecture generalizes the first conjecture because knotoids with their endpoints in the same region are equivalent to classical knots. We discuss the corresponding conjectures for Khovanov homology of knotoids.\\

Title : New Invariants of Links and Their State Sum Models

S.Lambropoulou and L.Kauffman

11 May 2017, 15:00-16:30 and 16:45-18:15 (Moscow time) at BMSTU.

Abstract : We introduce new 4-variable invariants of links, H[R], K[Q] and D[T], based on the invariants of knots, R, Q and T, denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. The new invariants are obtained by abstracting the skein relation of the corresponding invariant and making a new skein algorithm comprising two computational levels: first producing unlinked knotted components, then evaluating the resulting knots. We provide skein theoretic proofs of the well-definedness and topological properties of these invariants. State sum models are formulated and relationships with statistical mechanics models are articulated. Finally, we discuss physical situations where this course of action is taken naturally. The new invariants in this paper were revealed through generalizing the skein theoretic definition of the invariants Θd related to the Yokonuma-Hecke algebras and their 3-variable generalization Θ for classical links, which generalizes the Homflypt polynomial as well as the Gauss linking number.

The presentation will consist of two consecutive one-hour talks by the authors. Professor Lambropoulou will give the first talk, and Professor Kauffman will give the

second talk.

Title :  Knot invariant with multiple skein relations

Zhiqing Yang,

12 May 2017, 15:00-16:30 (Moscow time) at BMSTU.

Abstract : Given any oriented link diagram, one can construct knot invariants using skein relations. Usually such a skein relation contains three or four terms.The author introduces several new ways to smooth a crossings, and uses a system of skein equations to construct link invariant. This invariant can also be modified by writhe to get a more powerful invariant. The modified invariant is a generalization of both the HOMFLYPT polynomial and the two-variable Kauffman polynomial. Using the diamond lemma, a simplified version of the modified invariant is given. It is easy to compute and is a generalization of the two-variable Kauffman polynomial.

19 мая 2017 года в пятницу в 15:00(UTC+3):

Тема : «Doubled Khovanov homology»

William Rushworth

Abstract: Doubled Khovanov homology is an invariant of virtual links constructed by doubling up the chain groups of classical Khovanov homology. This allows for a new map to be incorporated, in addition to the multiplication and comultiplication familiar from the classical construction. Unlike other extensions of Khovanov homology to virtual links, the new map is non-zero; this allows the so-called 'problem face' - the primary difficulty in extending Khovanov homology to virtual links - to be dealt with in a new manner. Further, there is a perturbation of doubled Khovanov homology akin to that defined by Lee in the classical case, and a doubled Rasmussen invariant can be extracted from it. While this perturbed theory replicates the functorial behaviour of its classical counterpart, many connected cobordisms are assigned the zero-map, a consequence of the novel behaviour of the alternately coloured smoothings of virtual links.

In this talk we shall describe the construction of doubled Khovanov, and some of its applications. Further, we shall describe the perturbation and the definition of the doubled Rasmussen invariant, and show that concordances are assigned non-zero maps, using this to conclude that the odd writhe is an obstruction to sliceness.

Тема : «Что такое суперполином?»

С.Г.Гуков

Аннотация: Что касается аннотации, я постараюсь объяснить откуда берутся структурные свойства инвариантов узлов, а самое главное -- зачем они нужны, и чем могут быть полезны. Буду готов ответить на любые запросы аудитории и, в зависимости от интереса, могу сделать больше акцент либо на сами структурные свойства либо на топологическую теорию поля.

На кафедре организован научный семинар «Математическое моделирование процессов управления». Научный руководитель семинара - заведующий кафедрой, профессор, д.ф.-м.н., член-корреспондент РАН А.П. Крищенко.

Семинар ориентирован на студентов старших курсов, аспирантов и сотрудников кафедры. Заседания семинара проводятся по понедельникам в аудитории 1025 учебно-лабораторного корпуса МГТУ им. Н.Э. Баумана. Начало в 17:30.

Программа работы семинара

30 января 2012 года доклад О.И. Гончарова на тему: «Алгоритмы стабилизации билинейных систем».

23 января 2012 года доклад А.В. Наумова на тему: «Методы и алгоритмы решения задач стохастического линейного программирования с квантильным критерием».