2.6.2 ntangles and basic polyhedraEvery ntangle will be denoted by a regular 2ngon with 2n arcs emerging from its vertices. For every ntangle we can distinguish 2n possible positions obtained rotating the tangle by the angle [(pk)/n] (k=0,1,¼,2n1), and 2n positions of the tangle obtained by a mirrorreflection in a horizontal reflection line and then rotated by the angle [(pk)/n] (k=1,¼,2n1). For 2tangles there are two closures: numerator and denominator (N and D) closure, and for 3tangles there are two basic types of closures: a closure where only adjacent vertices are joined (Aclosure), and a closure where two opposite vertices are joined (Oclosure). Since the result of a closure depends on a position of 3tangle, for every 3tangle there are two possible Aclosures and three possible Oclosures. The number of closures, i.e., the number of ways of joining 2n points on a circle by n nonintersecting chords is known as Catalan number (or Segner number). For n £ 10 Catalan numbers (the sequence A000108 from OnLine Encyclopedia of Integer Sequences) are
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 ntangle (n ³ 3). An elementary ntangle with n1 vertices is denoted by n1 or 1 1¼ 1, where 1 occurs n1 times. As the basic position of elementary tangle we take the one where one strand is horizontal and remaining n1 strands are vertical. An elementary ntangle n1 induces a coordinate system of concentric regular 2ngons 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 ntangle placed in this coordinate system can be denoted by t_{1} t_{2} ¼t_{n1} (t_{n})_{r1} (t_{n+1})_{r2} ¼(t_{n+k1})_{rk}, where t_{i} (i=1,2,¼,n1) is an algebraic tangle placed in the corresponding vertex of n1 (in the order from the right to the left), and t_{j}_{rjn+1} (r_{jn+1} Î {1,¼,2n}, j=n,¼,n+k1, k=1,2,¼) is an algebraic tangle t_{j} placed in the region r_{jn+1}, between kth and (k+1)th concentric regular 2ngon (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 1_{rjn+1} we write just r_{jn+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.
