A knot K is a simple closed curve in 3-space. A generic projection of K on a plane results in a curve with a number of intersection points. The least such number cr(K) among all projections is called the crossing number of K. It is hard to compute in general, but there are several computable knot invariants that give lower bounds for it. I will describe two of them, the Jones polynomial and Khovanov homology, and state a few connections of these invariants with the crossing number.
Although the talk is about knots, no prior knowledge of knot theory will be assumed. For simplicity, most of the definitions and constructions will be done on pictures.