1.6 Classification of KLs
As we mentioned before, at the beginning we will not classify KLs according to a number of components. The number of components is an invariant that can be computed according to the Component Algorithm, but many other joint graph-theoretical or combinatorial properties of KLs can be discussed before introducing a number of components, or even without introducing it. For example, the most universal classification of KLs that contains two classes of KLs: 1*-links and polyhedral KLs is based on bigon collapse, without using a number of components.
We will consider only prime KLs, denoted by Conway notation. For both of them we will use the general term "link", considering knots as 1-component links. A knot or link shadow with single bigons is called a source link. Every source link induces an alternating KL, called source KL. Any KL can be derived from some source KL by replacing single bigons by chains of bigons. All links that could be derived from one source KL by such replacements form a family. KLs will be distributed into disjoint sets, named worlds by A. Caudron: algebraic and polyhedral, and their subworlds: rational, stellar, etc.
In general, all KLs can be
divided into two main classes: alternating and non-alternating ones, where KL is
alternating if it has an alternating representation. According to the result of
If we disregard handedness of KLs, every proper KL shadow uniquely defines an alternating KL. If bigons are denoted by colored (bold) lines, then among graphs obtained we distinguish regular and combined graphs: 3-regular (where in each 3-valent vertex there is exactly one colored edge), 4-regular, and combined 3- and 4-valent graphs. A link shadow is called basic if its edge-bicolored graph is regular. If it is 4-regular, such a graph is a basic polyhedron.
Two basic polyhedra or source links are equal iff they are isomorphic. For n ³ 12 different basic polyhedra or source links can be obtained as non-isomorphic projections of the same alternating KL. For the lower values of n it is not necessary to make distinction between basic polyhedra (source links) and their corresponding alternating KLs: there is no alternating KL that projects into different (non-isomorphic) basic polyhedra or source links with n £ 11 crossings.
It is interesting that the fundamental term "family" is very hard to find in knot theory books or papers. According to the book Knots and Surfaces (Farmer and Stanford, 1996), "a family of knots is an informal term used to describe a list of knots where each successive knot is obtained from the previous one by a simple process". The source link 2 (i.e. the Hopf link 212) and its family are shown in in the corresponding figure.