(Zoom) Семинары под руководством М.Э.Казаряна и С.К.Ландо «Комбинаторика инвариантов Васильева»: Сергей Ландо

In 1982, W. H. Cunnigham suggested a canonical decomposition of graphs into so-called prime graphs. For a connected graph with more than 5 vertices, this decomposition is unique, and it is closely related to the property of graphs to be intersection graphs of chord diagrams. In particular, a graph is an intersection graph if and only if each prime graph entering its canonical decomposition is one. In fact, if a prime graph is an intersection graph, then there is only one (up to reflection) chord diagram having it as the intersection graph, which allow the usage of Cunningham’s decomposition to describe all chord diagrams having a given graph as the intersection graph. Cunningham’s decomposition can be used to prove the following theorem (S.Chmutov, S.Lando, 2007) The value of the weight system associated to a Vassiliev knot invariant depends only on the intersection graph of a chord diagram rather than on the diagram itself if and only if this knot invariant cannot tell any knot from its mutant knot. (Mutation is an operation on knots, which consists in cutting off a ball from S3 intersecting the knot at 4 points and regluing it back after rotation.)

The main goal of the talk is approbation of techniques of distant seminar talks.