报告题目：Knot theory
报 告 人：Vassily Olegovich Manturov (Moscow Institute of Physics and Technology)
报告地点：ZOOM ID:818 3620 2175
https://us02web.zoom.us/j/81836202175
Lecture 1. Reidemeister moves. Colouring invariants, and the linking number.
2022/01/06 18:00-19:00
Abstract:We will introduce diagrams of knots and links. We use the diagrams
to build two invariants of links: Coloring invariant and the linking number.
Lecture 2. The Kauffman bracket, the Jones polynomial.
2022/01/09 18:00-19:00
Abstract:We will use Kauffman bracket to prove that Jones polynomial is a link invariant. We will compute some examples and prove the Kauffman-Murasugi-Thistlethwaite Theorem.
Lecture 3. Fundamental group. The knot group.
2022/01/13 18:00-19:00
Abstract:We will define a famous link invariant: the fundamental group of the knot complements. We will show that this group is not trivial if the knot is not trivial.
Lecture 4. The knot Quandle is a complete knot invariant.
2022/01/16 18:00-19:00
Abstract:Matveev and Joyce defined a knot invariant, the knot quandle. This is a complete invariant. We can obtain many invariants from the knot quandle, for example, the knot group and coloring invariant.
Lecture 5. The braid groups
2022/01/20 18:00-19:00
Abstract:We define the braid group. The action of the braid group on Aut(F_{n}) is complete.We will also discuss Markov Theorem and Alexander Theorem.
Lecture 6. The Alexander polynomial.
2022/01/23 18:00-19:00
Abstract:We will introduce several ways to define the Alexander polynomial.
Lecture 7. Vassiliev invariant
2022/01/27 18:00-19:00
Abstract:We will introduce the Vassiliev invariant and prove that “polynomial” invariant are all come from the Vassiliev invariant.
Lecture 8. Khovanov homology
2022/01/30 18:00-19:00
Abstract:Khovanov homology is a categorification of the Jones polynomial, in the sense that its Euler characteristic is the Jones polynomial.
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Introduction of the Speaker：Professor Vassily Olegovich Manturov is from Moscow Institute of Physics and Technology.His research interest is low dimensional topology and knot theory. He has published more than 150 papers and got more than 1500 citations. He got "Professor of RAS" in 2016 and he is one of the Managing Editors of the "Journal of Knot Theory and Its Ramifications". He has published many books, for instance, 《Parity in knot theory and graph-links. Contemporary Mathematics. Fundamental Directions》，《Low-dimensional Topology and Combinatorial Group Theory》，《Virtual Knots. The State of the Art》 and 《Knot Theory》. He held many international conferences, such as "4-th Russian China Russia-China on Knot theory and Related topics" and three International Conferences in the Mathematical Institute (Oberwolfach) on knot theory and low-dimensional topology".
