Statistical topology and knotting of fluctuating filaments

The aim of these notes is to provide an introduction to the topic of statistical topology. With this name we refer to a combination of ideas and techniques from statistical mechanics and knot theory used to study the entanglement properties of fluctuating filaments. Some questions that we are going to address are the following: (i) What is a knot and how can we identify it? (ii) Which is the probability of finding a random curve that is knotted? (iii) How complex are these knots? (iv) How big are they? To try to partially answer these questions we will make use of few paradigmatic problems and try to investigate them by providing some “state of the art” theoretical and numerical techniques. Exercises and lists of open problems will be provided too.