Classification of 2-bridge knots

In this thesis, we classify the $2$-bridge knots, the nontrivial knots which have a knot diagram with two local maxima and minima. They are in some sense the simplest type of knots after the trivial knot. To understand these knots, we focus on a particular representation of them, the Schubert normal form, which generates all the information about the structure of these knots, taking only two integers as an input. To complete the classification, we exploit the connection between knots and $3$-manifolds. We show that Lens spaces are the double branched coverings of the $3$-dimensional sphere branched along $2$-bridge knots, hence we get the classification of $2$-bridge knots using the classification of Lens spaces.