SCHWEITZER 2012 Problem 10

Let $K$ be a knot in the $3$ -dimensional space (that is a differentiable injection of a circle into $\mathbb{R}^3$ , and $D$ be the diagram of the knot (that is the projection of it to a plane so that in exception of the transversal double points it is also an injection of a circle). Let us color the complement of $D$ in black and color the diagram $D$ in a chessboard pattern, black and white so that zones which share an arc in common are coloured differently. Define $\Gamma_B(D)$ the black graph of the diagram, a graph which has vertices in the black areas and every two vertices which correspond to black areas which have a common point have an edge connecting them. [list=a] []Determine all knots having a diagram $D$ such that $\Gamma_B(D)$ has at most $3$ spanning trees. (Two knots are not considered distinct if they can be moved into each other with a one parameter set of the injection of the circle)[/] []Prove that for any knot and any diagram $D$ , $\Gamma_B(D)$ has an odd number of spanning trees.[/] [/list]