Probably the best explanation of Vogel's algorithm (1990), using the geopolitical language from the 1990s that may remain current till our own days, is given by A.Sossinsky (2002), so we paraphrase his description. Let us consider a planar map determined by a projection of a knot or link L. A country in this map is said to be in turmoil if it has two edges that belong to two different circles labelled with arrows going in the same direction. In the case of the knot 3 2, only the region T is in turmoil. An operation called perestroika can be applied to any country in turmoil. It consists in replacing two faulty edges by two "tongues", one of which passes over the other, forming two new crossings. The aim is to create a new digonal country (not in turmoil) and several new countries, some of which (in our example two of them) may be swallowed up by bordering countries. If after that there remain some nonnested Seifert circles, we apply the changeofinfinity operation to them. Vogel's algorithm is repeated as long as there are regions in turmoil and as long as there are some nonnested Seifert circles. Vogel's algorithm acting on the knot 3 2 is illustrated in the preceding figure. It is possible to use Vogel's algorithm to unroll a KL and obtain its braid representation. The Knot 2000 (K2K) function GetBraidRep (webMathematica GetBraidRep) is the implementation of Vogel's algorithm. Beginning from Pdata of an arbitrary KL diagram, by using Reidemeister's moves of type 2, that function transforms the parts of that diagram that satisfy certain conditions, until it gets the braid representation. As an external program it also uses the program Braid9.0 written by A. Bartholomew. In most cases, you can get a shorter braid word if you first reduce an input KL given by Pdata by using the function Reduction KnotLink, and then apply to that result the function GetBraidRep. To get a graphical output braid diagram for the braid word, one can use the function ShowBraid (webMathematica ShowBraid). The inverse function KnotFromBraid (webMathematica KnotFromBraid) produces Pdata from an arbitrary braid word.
