В. О. Мантуров (МФТИ, Москва)
A map from the classical world to the virtual world: how to use virtual knot theory techniques for classical objects.
Аннотация
Over the last 20 years virtual knot theory has experienced a very rapid development where many interesting structures arose. In particular, new invariants valued in pictures appeared. Hence, it becomes challenging to construct mappings from the "classical theory" to "virtual theory" (for example, in order to "pull back" invariants and structures arising in the latter).
In the talk we discuss various ways of constructing some maps. We start with a discussion of maps from classical braids to so-called "flat virtual braids" constructed in 2022 by Nikonov and the author. In the last part we discuss a way of looking at classical knots and braids from the point of view of horizontal secants instead of crossings and horizontal trisecants instead of crossings. Based on this, we construct maps from classical braids to virtual braids and the "secant quandle" (the latter is a joint work with Yangzhou Liu).
We pose a general question: how other "classical" objects can be mapped to their "virtual" counterparts.
