Knots

This thesis is the result of my wish to deepen the study of topology, introduced to me in the course "Complementi di Matematica", taught by professor Salamon during the second year of the laurea degree course in Matematica per le Scienze dell'Ingegneria at the Politecnico. The work is about the mathematical and topological theory of knots and its connections to the theory of surfaces.

In the first chapter I present the results concerning planar diagrams of knots, classifying them through the equivalence relation of isotopy. Some numerical properties of these diagrams are also defined.

The second chapter is about the various polynomials associated to knots; these will prove to be a very strong instrument of analysis: we'll be able to link any knot to a particular polynomial, thus turning the problem of identifying isotopic knots into a simple computation.

The results connecting knots theory and surfaces theory are presented in the third chapter, after a mathematical formalization of the concepts used at that point and an introduction to the topological method of surfaces analysis; with those instruments we'll prove the major theorem about knots and surfaces, saying that any knot is the boundary curve of an orientable surface; I was able to show here the surfaces constructed in the demonstration thanks to SeifertView, a free software developed by Jarke J. van Wijk and Arjeh M. Cohen, available online.

In the appendix I present some ruled surfaces I constructed from the so-called torus knots using Wolfram Mathematica with the help of professor Salamon.