Can’t Occur as a Knot Projection: 4-Valent Multi-3-gon 3-Polytopes

When one is trying to understand a “mathematical landscape” it is often helpful to construct lots of examples and collect data about these examples as a way of being able to demonstrate new facts about this mathematical landscape. If one has a knot in 3‑dimensional space it is always possible to choose a direction in space, and project this knot into a plane so that one gets a diagram consisting of dots and lines (called a graph) which represents the knot and where the “crossings” of the knot appear in the projection as dots where there are 4 line segments at each dot. In this representation of the knot we disregard the under and over information about the strands of the knot. Thus, we are lead to a graph theory situation, drawings of dots (vertices) and lines, which might be straight or curved (edges) in the plane where there are 4 lines (edges) at every dot (vertex). Verify for yourself that the graph in Figure 1 can be interpreted as the projection of a knot with 9 crossings.