Knots, Diagrams and Kids’ Shoelaces. On Space and their Forms

Knots and links are mathematical objects living in our three-dimensional space or in other ambient spaces, such as the 3-sphere and the four-dimensional space, and they possess a rich variety of properties and structures, some of them ore quite elementary while some other are very complex and difficult to grasp. Knot theory has extensive interactions, not only with different branches of mathematics, but also with various and fundamental areas of physics. Knots and links are deeply related to the geometry of 3-manifolds and low-dimensional topology, quantum field theory and fluid mechanics. We survey some current topics in the mathematical theory of knots and some of their more striking ramifications in physics and biology. The study of knots is essential to the comprehension of three- and four-dimensional spaces. Knot theory is of central importance in mathematics, as it stands at a crossroad of topology, geometry, combinatoric, algebra and mathematical physics. And it is likewise a key ingredient entering in the comprehension of dynamical systems, macroscopic physics and biochemistry. There are essentially four mathematical approaches to the study of knots and their polynomial invariants, combinatorial, as in the Alexander and Conway case, geometrical, via constructions called Seifert surfaces, algebraic, by considering the group of the knot, and physical, developed particularly by Witten, in which the Jones polynomial is interpreted and generalized using Chern-Simons theory. This article aims at stressing the importance of considering diagrams in the study of topological and geometrical objects and the key role of knot theory for the understanding of the structure of space and space-time.