In this page you can find marerial related to the Seminar

"Knot theory & Low-Dimensional Topology"

that took place at China Agricultural University; College of Science, September & October 2019.

1st Announcement; Seminar

"Knot Theory

&

Low-Dimensional Topology"

各位同学，下学期国际学院 Yannis 教授希望在我们系开办纽结理论讨论班，介绍流形的拓扑，纽结理论初步，和代数拓扑等概念，讨论班每周1小时，计划讲 9 到 12 次。纽结理论与数学、物理、拓扑等密切相关，是非常有趣的数学分支，讨论班结束后我们会办法结课证书。有意向参加的同学可以先填写这个统计问卷：https://www.wjx.top/jq/41150833.aspx

Dear students, Professor Yannis Diamantis of the International College will hold a seminar on knot theory in our department (College of Science) next semester. He will introduce the concepts of manifold topology, knot theory, and algebraic topology. Discuss classes will be 1 hour long per week and plan to cover 9 times. The knot theory is closely related to mathematics, physics, topology and so on. It is a very interesting branch of mathematics. When the mini course is finished, course certificates will be given to students that attended. Interested students can fill out this statistical questionnaire at

https://www.wjx.top/jq/41150833.aspx

Abstract

In the last three decades we have witnessed the growth of a fascinating mathematical theory called Knot Theory. Knot theory is a subeld of Topology, the study of properties of geometric objects that are preserved under deformations, and an advantage of knot theory over many other fields of mathematics is that much of the theory can be explained at an elementary level. The first part of the seminar is devoted to a short introduction to knot theory in a rapid penetration into key ideas and examples. We will present all essential background material of classical theory of knots, links and braids in S^3 culminating to the Jones polynomial, that we will define using the Kauffman bracket polynomial and via the Tempreley-Lieb algebra. We will then present a knot theoretic approach to the quantum group SL(2)_q via the Kauffman bracket polynomial. This way we also find a solution to the famous Yang-Baxter equation. We will then generalize the Jones polynomial to other 3-manifolds (skein modules). In particular, we will first introduce the notion of 3-manifolds, Dehn surgery, which is a simple way to obtain all 3-manifolds from S^3 and Kirby calculus, namely, an equivalence relation between (framed) knots that represent homeomorphic 3-manifolds. We will then generalize the Jones polynomial for knots and links in the Solid Torus, the Lens spaces L(p,q) and Homology Spheres. Our motivation is to construct 3-manifold invariants via quantum groups and our aim is to obtain a uniform algebraic approach to these (Witten) invariants, which remains an open problem in Low-Dimensional-Topology.