Have you ever dreamed of drawing a picture, moving things around with your hands, and claim it is a proof? Well, I have. Graphical calculus develops a theory of what formally represents a drawing, and what you are allowed to do with your drawings without changing the object it represents. We will go through Reshetikhin--Turaev theory of quantum link invariants, following in parallel the theory for drawings moves (graphical calculus), the category of objects they represent (ribbon categories), and algebraic descriptions of these (quantum groups).
