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
complete bigon collapse, alternating KL can be divided into two main worlds: algebraic and polyhedral, but even this depends from the definition of an algebraic KL.

There are two possibilities for defining algebraic KLs:

          1) a KL is algebraic if it has a minimal algebraic representation;
          2) a KL$ is algebraic if it has an algebraic representation.

For example, alternating link .3.2.3.2 with n=12 crossings has algebraic representation (4,-3 1) (3,-2 1) with n=14 crossings, so it will be treated as polyhedral according to the first, and as algebraic according to the second definition. Here and in the sequel we will accept the first definition, so it will be treated as polyhedral.

With non-alternating KL situation is even more complicated: without introducing additional criteria, we are even not able to distinguish algebraic non-alternating KLs from polyhedral ones, because the same non-alternating KL can have an algebraic and a polyhedral representation that are both minimal. For example, the same non-alternating link can be represented as (4,-2) (2,2), or 2 0.-2.-2 0.-2 0. According to the definition of algebraic KLs that we accepted, the link in question will be algebraic, despite of that it has the polyhedral minimal representation as well.

Depending from the type of their corresponding graphs (Caudron, 1982), i.e., depending from the tangle operations used for their derivation, algebraic KLs can be rational, stellar, or
arborescent. Rational tangles and rational KLs obtained as their closures are obtained by using the operation of product, and stellar (pretzel) KLs are obtained composing rational tangles by the operation of ramification. Finally, composing rational and stellar tangles by the operations of product and ramification, arborescent KLs are obtained, so we have the complete
stratification of algebraic alternating KLs. Unfortunately, in the polyhedral world it is not possible to establish the complete order, because we cannot establish the complete order of basic
polyhedra. For example, there are two different (non-isomorphic) basic polyhedra 136* and 1318* with n=13 crossings that are two non-isomorphic diagrams of the same alternating link. Some perspective for a future classification of basic polyhedra offers Crazy Spider Algorithm, giving possibility to derive families of basic polyhedra, establish their generic notation and the order based on the family principle.

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. 


 
 

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