学术报告(Grigory Mikhalkin 1.15)

发布人:周妍 发布日期:2019-01-03
活动时间
-
活动地址
新数学楼 415
主讲人
Grigory Mikhalkin 教授(Universite de Geneve)

题目:Maximally writhed real algebraic knots and links

摘要: The Alexander-Briggs tabulation of knots in R3 (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 RP3 rather than R3, and use a different number also serving as a measure of complexity of a knot: the minimal degree of a real algebraic curve representing this knot. As it was noticed by Oleg Viro about 20 years ago, the writhe of a knot diagram becomes an invariant of knots in the real algebraic set-up, and corresponds to a Vassiliev invariant of degree 1. In the talk we’ll 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 RP3 for every degree d. Furthermore, this real algebraic knot type has a number of other 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.