Before we begin derivation of KLs belonging to other worlds: stellar (or prismatic), arborescent stellar, arborescent generalized, and polyhedral, we will consider some invariants, mostly polynomial, by which we will be able to distinguish and recognize KLs. 

R. Fox defined very elementary invariant of KLs, that we call now Fox 3-coloring (Crowel and Fox, 1963). In an oriented KL we begin by labelling each oriented arc connecting two successive undercrossings by xi ( i = 1,2,...,m). These labelled arcs will be the generators of the KL. To each generator we can assign one color from a set of 3 colors. A knot or link is 3-colorable if in every crossing at which two colors appear, three different colors appear. In other words, in every vertex all generators can have the same color, or they all have different colors. If a link L has at least one 3-colorable diagram, then each of its diagrams is 3-colorable. From this follows that the 3-colorability of a link diagram is the link invariant. Three-colorability was one of the first methods used in knot theory to distinguish knots. For example, a trefoil can be distinguished from a figure-eight knot, because the first is three-colorable, and the other is not.

The 3-coloring can be generalized to k-coloring (k>3). A link diagram is k-colored if every generator is labeled by one of the numbers 1,2,...,k in such a way that at each crossing the sum of the labels of the undercrossings (incoming and outgoing generator) is equal to twice the label of the overcrossing (passing generator) modulo k. In other words, if iV, oV, and pV are the labels of the incoming, outgoing, and passing generator in the crossing $V$, then in every crossing holds the relation iV+oV=2pV (mod k). It holds the

Labeling theorem: if some diagram of a KL can be labeled mod k, then every diagram of that KL can be labeled mod k (Livingston, 1993; Przytycki, 2004).

A link L can be colorable with regard to several different values of k. The set of different numbers of colors with which L may be colored is the coloring number set of L.  Fox k-colorings are special cases of quandles introduced by Joyce and Matveev (Joyce, 1982; Matveev, 1982). The detailed discussion of quandles is given in the book Knot Theory (Chapter 5) by V. Manturov (2004).

A KL projection is perfectly colorable if three different colors appear in every vertex.

KL given by its Conway symbol, Dowker code, or P-data, is an input for the LinKnot function fGenerators (webMathematica fGenerators), which computes generators of the KL with the list of corresponding signs of crossing points. The result is a list of ordered triples containing incoming, outgoing, and passing generator (I O P= Incoming-Outgoing-Passing) for each vertex, divided according to the components of the KL. 

The function fColTest (webMathematica fColTest) has the same input as fGenerators with the additional number n ( n ³ 3) denoting how many colors should be used for KL projection coloring. The result is the list of generators, list of their labeling, and the list of generator colors. The result can be a perfect coloring, or standard coloring. For example, the link 2,2,2 (or 613) is k-colorable if k=0(mod 2) or k=0(mod 3), so its color set is {2,3,4,6,8,9,10,12,14,15,...}, where the perfect colorings are obtained for k=0(mod 3).