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A reader familiar with BFRs can recognize a similarity of coordinate basic polyhedra notation and their corresponding minimal braid words:
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=3,4,¼),
given by the minimal code |2| (1 2)n-1,
where (1 2)n-1
stands for 1 2 ¼1 2, with
1 2 repeated n-1 times. Every other minimal code of a basic polyhedron derived
from a n-tangle (n ³ 3) is
of the form |n-1| s0k s, where s0k
is an alternating sequence 1 2 ¼
of the length k (k ³ 1),
and s is a sequence of numbers ri (ri Î {1,2,¼,2n},
i=1,2,¼) 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
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
³ 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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