A reader familiar with BFRs can recognize a similarity of coordinate basic polyhedra notation and their corresponding minimal braid words:

6* (Ab)3
8* (Ab)4
9* AbACbACbC
10* (Ab)5
10** AbAbCbACbC


The main difference is that Crazy Spider Algorithm can be used for the exhaustive derivation of basic polyhedra resulting in the corresponding notation, while BFRs can be used only to recognize certain families of basic polyhedra, without providing an algorithm for their exhaustive derivation.

Composite basic polyhedra (starting from 10***) can be also represented as the compositions of non-algebraic tangles. The first of them, 10*** is 5* 5*, the product of two hyperbolic tangles |2| 1 2 1. In the same way, 11*** is 5* 1 5*, 12E is 5*,1,5*,1=5* 2 5* ~ 11***2, 12I is 7* 5*, 12J is 5* 1 1 5*, etc.

Unfortunately, codes denoting basic polyhedra are not unique. Their number grows very fast as the number of crossings increases.

For the minimization of codes and reduction of their number we use commutativity of vertices belonging to non-adjacent regions, with the requirement that no bigons are created.

For example, the code |3| 1 2 3 5 4 5 2 1 2 is minimized to |3| 1 2 3 2 1 2 5 4 5. Instead of minimizing all possible codes according to rules mentioned, it is possible to construct in advance all minimized codes, and then choose the minimal among them.

LinKnot function fBasicTan (webMathematica fBasicTan) gives all minimized closed n-tangles with k crossings, and the function fMakeBP (webMathematica fMakeBP) for given n from the list of minimal n-tangle codes makes all basic polyhedra with 1,2 sequence of the length k, minimal n-tangle code from the position s, and 1,2 or 2,1 sequence of the length l. The function fMakeAllnsBP (webMathematica fMakeAllnsBP) for given n, from the list of minimal n-tangle codes makes all basic polyhedra with minimal n-tangle code from the position s, and with c crossings.

Now we will consider families of minimal representations of basic polyhedra. The first family consists of n-antiprismatic basic polyhedra (2n)* (n=3,4,), given by the minimal code |2| (1 2)n-1, where (1 2)n-1 stands for 1 2 1 2, with 1 2 repeated n-1 times. Every other minimal code of a basic polyhedron derived from a n-tangle (n 3) is of the form |n-1| s0k s, where s0k is an alternating sequence 1 2  of the length k (k 1), and s is a sequence of numbers ri (ri {1,2,,2n}, i=1,2,) denoting regions, not beginning with 1 or 2. A family of basic polyhedra derived from s consists of all basic polyhedra of the form |n-1| s0k s obtained for a fixed s, and can be denoted by (n,s,k). Basic polyhedra belonging to a (n,s,k)-family for fixed n and s are obtained by the same closure. For example, for s=3 2 1 2 and k 3 it is obtained the family |2| s0k s, that consists from the basic polyhedra 9*, 10**, 11*, 12B,... given by the minimal codes |2| 1 2 1 3 2 1 2, |2| 1 2 1 2 3 2 1 2, |2| 1 2 1 2 1 3 2 1 2, |2| 1 2 1 2 1 2 3 2 1 2,...; for s=6 1 2 1 2 1 and k 1 it is obtained the family |2| s02k+1 s, beginning with the basic polyhedron 11** given by the code |2| 1 2 1 6 1 2 1 2 1, etc. The series s0k is the k-antiprismatic belt of the basic polyhedron, and s is a 3-tangle. Some properties of basic polyhedra are not only properties of individual basic polyhedra, but can be extended to whole families. For example, in the family |2| s0k s beginning with 9* every basic polyhedron obtained for k=1 mod 3 is a two-component link, and knot otherwise.
 

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