Applications of combinatorics to knot theory

If we project a knot to a plane we obtain a 2-dimensional curve, a shadow of the knot. Except for very special cases, a shadow does not determine a knot: in general, more than one knot can be projected to a given shadow. The relevant question is: which properties of a knot can be obtained from its shadow? In this talk we will illustrate how standard techniques from topological and extremal graph theory can be used to investigate this problem. No previous knot theory knowledge is expected from the audience (and very little knot theory knowledge should be expected from the speaker).