Knot Theory - Knot Diagrams

Knot Diagrams

A useful way to visualise and manipulate knots is to project the knot onto a plane—think of the knot casting a shadow on the wall. A small change in the direction of projection will ensure that it is one-to-one except at the double points, called crossings, where the "shadow" of the knot crosses itself once transversely (Rolfsen 1976). At each crossing, to be able to recreate the original knot, the over-strand must be distinguished from the under-strand. This is often done by creating a break in the strand going underneath. The resulting diagram is an immersed plane curve with the additional data of which strand is over and which is under at each crossing. (These diagrams are called knot diagrams when they represent a knot and link diagrams when they represent a link.) Analogously, knotted surfaces in 4-space can be related to immersed surfaces in 3-space.