MathDB

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.
[*]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.[/*]

Problem Statement