Statistical topology of closed curves: Some applications in polymer physics

Topological entanglement in polymers and biopolymers is a topic that involves different fields of science such as chemistry, biology, physics, and mathematics. One of the main issues in this topic is to understand how the entanglement complexity can depend on factors such as the degree of polymerization, the quality of the solvent, and the temperature or the degree of confinement of the macromolecule. In this respect a statistical approach to the problem is natural and in the last few years there has been a lot of work on the study of the entanglement complexity of polymers within the statistical mechanics framework. A review on this topic is given here stressing the main results obtained and describing the tools most used with this approach.