Let K be a knot in the 3-dimensional space (that is a differentiable injection of a circle into R3, 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 ΓB(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.[*]Determine all knots having a diagram D such that ΓB(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, ΓB(D) has an odd number of spanning trees.[/*] college contestsMiklos Schweitzer