Links modulo knots: 1. The linking number and the triple $\bar\mu$-invariant

This is the first lecture of a mini-course which is a part of the Euler's Institute thematic program “New Trends in Topology”.

Abstract of the mini-course: In classical knot theory, links are often treated as “multicomponent knots” and play a largely auxiliary or secondary role with respect to genuine knots. But there also exists another line of research, which is concerned only with interaction of distinct components of a link and ignores local knotting of its individual components. This subject of “links modulo knots” has been actively developing in the last 65 years and can be more specifically described as studying equivalence relations on links such as PL isotopy (where local knots may appear and disappear on each component), link homotopy (where each component may intersect itself, but distinct components may not intersect each other), topological isotopy (i.e. homotopy through topological embeddings) and some other ones. This approach leads to a mathematics that is quite different from the usual knot theory. In some respects it is easier: a lot can be achieved by using only classical algebraic topology (homology, fundamental group, homotopy groups of spheres, etc.) which cannot be said of the usual knot theory. In some respects it is harder: the role played by the “multicomponent knots” in classical knot theory is taken here by links colored in $n$ colors (where the “multicomponent knots” correspond to the case $n=1$); because of this, one has to deal with $n$ variables instead of one variable, and to be prepared that in some settings they will not commute. The mini-course will be devoted to classical and modern constructions and results.

Disclaimer: In view of Article 28 of the Constitution of Russia, the fact that I'm teaching this course does not imply my automatic agreement with all actions of the Russian state, nor that I automatically recognize all federal elections in Russia as fair.

Zoom: https://mi-ras-ru.zoom.us/j/91599052030
Access code: the Euler characteristic of the wedge of two circles
(the password is not the specified phrase but the number that it determines)