Maximally writhed real algebraic knots and links

Date
2018/06/06 Wed 16:30 - 17:30
Room
3号館110講演室
Speaker
Grigory Mikhalkin
Affiliation
University of Geneva
Abstract

The Alexander-Briggs tabulation of knots in $\mathbb{R}^3$ (started almost a century ago, and considered as one of the most traditional ones in classical Knot Theory) is based on the minimal number of crossings for a knot diagram. From the point of view of Real Algebraic Geometry it is more natural to consider knots in $\mathbb{RP}^3$ rather than $\mathbb{R}^3$, and also to use a different number also serving as a measure of complexity of a knot: the minimal degree of a real algebraic curve representing the knot.

As it was noticed by Oleg Viro about 20 years ago, the writhe of a knot diagram becomes an invariant of a knot in the real algebraic set-up, and corresponds to a Vassiliev invariant of degree 1. In the talk we’ll briefly survey these notions, and consider the knots with the maximal possible writhe for its degree. Surprisingly, it turns out that there is a unique maximally writhed knot in $\mathbb{RP}^3$ for every degree d. Furthermore, this real algebraic knot type has a number of characteristic properties, from the minimal number of diagram crossing points (equal to $d(d-3)/2$) to the minimal number of transverse intersections with a plane (equal to $d-2$). Based on a series of joint works with Stepan Orevkov.