Diagrammatic approach to the classification of knots and contact topology

• 主讲人：Ivan Dynnikov (Steklov Mathematical Institute of RAS)
• 举办方： Beijing-Moscow Mathematics Colloquium
• 时间： 2023-01-13 17:00 - 2023-01-13 18:00
• 地点： online

Recording: https://disk.pku.edu.cn:443/link/A07BDF7E34AC9C1BD87EB37BF6B9E789
Valid Until: 2027-02-28 23:59

 

Abstract: About two decades ago I discovered that the problem of recognizing the unknot could be solved using a monotonic simplification procedure for certain type knot diagrams, which I called rectangular. They are also now known as grid diagrams. For arbitrary knots, the procedure does not work as well as it does for the unknot, since there may be many non-equivalent rectangular diagrams representing the same knot type that are not subject to any simplification by elementary moves. However, for many knot types, the set of non-simplifiable diagrams is finite, and it does not seem impossible that this is true for all knots. The classification of non-simplifiable rectangular diagrams turns out to be deeply related to the classification of Legendrian knots. This relation has been recently studied in our joint works with Maxim Prasolov and Vladimir Shastin. As a by-product of this study, we obtained an algorithm for comparing Legendrian knots, which has not been known before. In my talk, I will overview the most recent results on the subject.

The present work is supported by the Russian Science Foundation under grant~22-11-00299.

 

Bio: Ivan A. Dynnikov is a Professor at the Department of Higher Geometry and Topology, Lomonosov Moscow State University and also a Leading Scientific Researcher at Steklov Mathematical Institute of RAS. He obtained his PhD from Lomonosov Moscow State University in 1996 and became Doctor of Sciences in 2007. Dynnikov's research interests include low-dimensional topology and its applications in mathematical physics, knot theory, discrete Schrödinger operators, and the theory of dynamical systems. He was an invited speaker at the 2nd European Congress of Mathematicians (Budapest, 1996). In 2000, I.A. Dynnikov was awarded the prize of the Moscow Mathematical Society for young scientists. I.A. Dynnikov is the Scientific Secretary of the Moscow Mathematical Society and the author of more than 35 scientific publications.