The study of knots began in the 19th century with some mathematicians such as Gauss, Listing and Tait. In 1898, Tait writes in a preface of his work:

The subject [knot theory] is a very much more difficult and intricate one than at first sight one is inclined to think, and I feel that I have not succeeded in catching the key-note.

Until now, many mathematicians have been working on the study of knots, and no key-note has been found, or maybe there is no “key-note”. This shows that knot theory is a much deeper area of research than the first impression.

In this talk we will introduce and discuss some basic motivations behind the study of knots and some applications to dynamical systems for the classification of flows on three-dimensional manifolds.