User:Chan-Ho Suh/todo/draft3

From Wikipedia, the free encyclopedia

In the mathematical field of knot theory, a knot diagram refers to a particular way of representing a tame knot in R^3. Such a diagram is defined to be a connected finite graph on the plane. Each vertex has four edges and each pair of nonadjacent edges is considered to be a strand. One strand is marked over and the other is marked under. These choices for over and under are usually indicated by creating breaks between the vertex and the edges marked under.

A knot is given a diagrammatic representative by projecting it onto a plane; the choice of plane is important, as a careless choice could give a diagram that is not 4-valent, but of higher valence. Additionally, other undesired possibilities, such as tangencies, may occur. A slight perturbation of any plane will give a nice projection resulting in only double point singularities. This enables the marking of over and under as mentioned previously.

Note, however, that because of our choice of plane, a knot may have many very different appearing diagrams. For example, a knot may have an alternating projection

Possible topics: Unknot diagrams invariants of diagrams reidemeister moves