DescriptionIn 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. 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.
|Period||9 Oct 2010 → 29 Oct 2019|