2.6.3 Non-algebraic tangle compositions and component algebra

In the set of n-tangles we introduce various operations under the common name- compositions.

A composition of two n-tangles is a n-tangle 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 n-tangles (n ³ 2) is closed under compositions. For example, in the set of 2-tangles we have three operations: sum, product, and ramification resulting in a new 2-tangle.

The concept of tangle composition can be extended to the set of tangles that consists from n1-, n2-,...,nm-tangles (nm 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 2m-tangles is closed under tangle composition). We will consider only 2-tangles and 3-tangles.

The graph consisting of a regular 2n-gon and n chords joining its distinct vertices is called a chord diagram of order n, or shortly n-diagram. Let the symmetry group G act on a chord diagram. Two n-diagrams are equivalent iff there exist an element of the group G that transforms one to another. The set of n-diagrams quotient by the equivalence relation coming from the action of identity group G will be called the complete set of n-diagrams, or the set of positions of n-diagrams. If G is the dihedral group G=Dn of the order 2n we get the set of basic n-diagrams.

Main goal of this section is to determine number of components of KLs obtained as a closure of a composition of n-diagrams.

First let us introduce n-tangle types. In every n-tangle vertices can be substituted by algebraic tangles. There are three basic types of algebraic tangles, k, k, and [¥]k, where k is the number of internal closed components.

From every n-tangle we obtain its corresponding n-diagrams (or Gauss n-diagrams). 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).

 n 3 4 5 6 7 8 9 10 11 5 17 79 554 5283 65346 966156 16411700 3127002217

The five diagrams obtained for n=3 and 17 diagrams obtained for n=4 are illustrated in the following figure and denoted, respectively, by 3.1-3.5 and 4.1-4.17.