Parity in Knot Theory

活动时间： 04月29日15时00分       

地            点  ： ZOOM：869 7018 3616 密码：505895

讲座内容：

The aim of the talk is to describe thenotion of parity which appears in knot theory and low-dimensional topology anddiscriminates between even and odd crossings (nodes, double lines etc) ofvarious diagrams of knotted objects. Free knots are a drastic simplification ofvirtual knots which is highly non-trivial. Parity theory allows one toconstruct the parity bracket, a diagram-valued invariant of knotted objectwhich leads to the following principle: If a diagram is complicated enough(say, odd and irreducible) then it realises itself (i.e., appears as asubdiagram in a certain sense) in any diagram equivalent to it. This yieldsminimality in a very strong sense: not only we know the minimal complexity of alldiagrams of a given class, but also we know that all of them contain the uniqueminimal representative as a subdiagram (like words in the free group containingits minimal representative).We shall also touch on the application of paritytheory in cobordisms of knotted objects (free knots, virtual knots) and, inparticular prove the theorem that“If a free knot isslice, totally odd and irreducible then it is elementarily slice.”Thismeans that if some graph can be capped by a folded discs with typicalsingularities (double lines, triple points, cusps) then if all crossings areodd then it can be capped by a disc only with double lines. This makes thechecking procedure completely handy.

主讲人介绍：

Professor Vassily Olegovich Manturov is from MoscowInstitute of Physics and Technology.   Hisresearch interest is low dimensional topology and knot theory.   He has published more than 150 papers and gotmore than 1500 citations. He got "Professor of RAS" in 2016 and he isone of the Managing Editors of the "Journal of Knot Theory and ItsRamifications". He has published many books, for instance, 《Parity in knot theory and graph-links.Contemporary Mathematics. Fundamental Directions》， 《Low-dimensional Topology and Combinatorial GroupTheory》， 《Virtual Knots. The State of the Art》 and   《Knot Theory》.   He held many internationalconferences, such as "4-th Russian China Russia-China on Knot theory andRelated topics" and three International Conferences in the MathematicalInstitute (Oberwolfach) on knot theory and low-dimensional topology".

发布时间：2022-04-23 21:13:22