2.6.3 Nonalgebraic tangle compositions and component algebra In the set of ntangles we introduce various operations under the common name compositions. A composition of two ntangles is a ntangle obtained by joining in pairs n adjacent arcs emerging from the first tangle with n adjacent arcs emerging from the other. Note that the set of all ntangles (n ³ 2) is closed under compositions. For example, in the set of 2tangles we have three operations: sum, product, and ramification resulting in a new 2tangle. The concept of tangle composition can be extended to the set of tangles that consists from n_{1}, n_{2},...,n_{m}tangles (n_{m} ³ 2, m Î {1,2,¼k}), where the number of joined arcs is chosen in such a way that every tangle obtained by a composition has 2m free arcs (i.e. so that a set of 2mtangles is closed under tangle composition). We will consider only 2tangles and 3tangles. The graph consisting of a regular 2ngon and n chords joining its distinct vertices is called a chord diagram of order n, or shortly ndiagram. Let the symmetry group G act on a chord diagram. Two ndiagrams are equivalent iff there exist an element of the group G that transforms one to another. The set of ndiagrams quotient by the equivalence relation coming from the action of identity group G will be called the complete set of ndiagrams, or the set of positions of ndiagrams. If G is the dihedral group G=D_{n} of the order 2n we get the set of basic ndiagrams. Main goal of this section is to determine number of components of KLs obtained as a closure of a composition of ndiagrams. First let us introduce ntangle types. In every ntangle vertices can be substituted by algebraic tangles. There are three basic types of algebraic tangles, [1]_{k}, [0]_{k}, and [¥]_{k}, where k is the number of internal closed components. From every ntangle we obtain its corresponding ndiagrams (or Gauss ndiagrams). The number of chord diagrams can be computed by combinatorial methods. The number of the basic chord diagrams for n=3,¼11 is given in the following table (Khruzin, 2000).
The five diagrams obtained for n=3 and 17 diagrams obtained for n=4 are illustrated in the following figure and denoted, respectively, by 3.13.5 and 4.14.17.
