Looking at bent wires – -codimension and the vanishing topology of parametrized curve singularities

Abstract

Projecting a knot onto a plane – or, equivalently, looking at it through one eye – one sees a more or less complicated plane curve with a number of crossings (‘nodes’); viewing it from certain positions, some other more complicated singularities appear. If one spends a little time experimenting, looking at the knot from different points of view, then provided the knot is generic, one can convince oneself that there is only a rather short list of essentially distinct local pictures (singularities) – see Fig. 3 below. All singularities other than nodes are unstable: by moving one's eye slightly, one can make them break up into nodes. For each type X the following two numbers can easily be determined experimentally:1. the codimension in ℝ3 of the set View(X) of centres of projection (viewpoints) for which a singularity of type X appears, and2. the maximum number of nodes n into which the singularity X splits when the centre of projection is moved.