2.6.2  n-tangles and basic polyhedra

Every n-tangle will be denoted by a regular 2n-gon with 2n arcs emerging from its vertices. For every n-tangle we can distinguish 2n possible positions obtained rotating the tangle by the angle [(pk)/n] (k=0,1,,2n-1), and 2n positions of the tangle obtained by a mirror-reflection in a horizontal reflection line and then rotated by the angle [(pk)/n] (k=1,,2n-1).

A closure of n-tangle is obtained by joining in pairs remaining free arcs without crossing.

For 2-tangles there are two closures: numerator and denominator (N and D) closure, and for 3-tangles there are two basic types of closures: a closure where only adjacent vertices are joined (A-closure), and a closure where two opposite vertices are joined (O-closure).

Since the result of a closure depends on a position of 3-tangle, for every 3-tangle there are two possible A-closures and three possible O-closures. The number of closures, i.e., the number of ways of joining 2n points on a circle by n non-intersecting chords is known as Catalan number (or Segner number). For n 10 Catalan numbers (the sequence A000108 from On-Line Encyclopedia of Integer Sequences) are

n 2 3 4 5 6 7 9 9 10
  2 5 14 42 132 429 1430 4862 16796


In general, Catalan number is given by the formula C(n)=2n!/n!(n+1)!. LinKnot function fAllClosures (webMathematica fAllClosures) gives the list of all closures of a n-tangle (n
3).

An elementary n-tangle with n-1 vertices is denoted by |n-1| or |1 1 1|, where 1 occurs n-1 times. As the basic position of elementary tangle we take the one where one strand is horizontal and remaining n-1 strands are vertical. An elementary n-tangle |n-1| induces a coordinate system of concentric regular 2n-gons and corresponding regions, where the first lower middle or right region with two vertices is denoted by 1, and other regions (from 1 to 2n) are given in a clockwise order. Every n-tangle placed in this coordinate system can be denoted by |t1 t2 tn-1| (tn)r1 (tn+1)r2 (tn+k-1)rk, where ti (i=1,2,,n-1) is an algebraic tangle placed in the corresponding vertex of |n-1| (in the order from the right to the left), and tjrj-n+1 (rj-n+1 {1,,2n}, j=n,,n+k-1, k=1,2,) is an algebraic tangle tj placed in the region rj-n+1, between kth and (k+1)th concentric regular 2n-gon (at the kth level). Since our primary interest is derivation of basic polyhedra we start with algebraic tangles 1 and ensure that we create no bigons. Therefore, all pairs of adjacent regions must have different indexes. If all algebraic tangles are 1, in order to simplify notation, instead of 1rj-n+1 we write just rj-n+1.

In the initial state, all (potential) algebraic tangles have the same orientation. In our notation the symbol 0 have the same meaning as in the Conway notation for polyhedral KLs.

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