The number of components of a basic polyhedron is not the property of a single basic polyhedron, but of the whole family and depends from family parameters.

The same holds for some other KL invariants, like signature or the unknotting (unlinking) number. This can be illustrated on the example of the one-parameter family of basic polyhedra |2| s0k s (k ³ 3, s=3 2 1 2), starting with the basic polyhedron 9*. For k=3,¼,21 we obtained the following table.

For k=3,¼,21, the second column gives the ordering number of the basic polyhedron from the LinKnot data base (for k ³ 20), the third column the number of components, the fourth column the signature, and the fifth column the unknotting (unlinking) number of the basic polyhedron computed according to Bernhard-Jablan Conjecture (Jablan and Sazdanovic, 2005a).

k BP s u
3 9* 2 2
4 10** 2 1
5 11* 1 0
6 12B 1 0
7 133* 3 1
8 148* 2 0
9 1510* 3 2
10 1625* 3 1
11 17455* 2 0
12 182675* 2 0
13 195031* 3 1
14 2031002* 3 0
15   3 2
16   4 1
17   3 0
18   3 0
19   4 1
20   4 0
21   4 2

Following patterns appear in the table above:

  • the number of components is 2 for k=1 mod 3, and 1 otherwise;
  • the signature is 1 for k=1 mod 3, 0 for k=2 mod 3 or k=0 mod 6, and 2 otherwise;
  • the unknotting (unlinking) number is k for k=0 mod 6, k+1 for k=2 mod 6 or k=5 mod 6, and k+2 otherwise.

Connections between the families of basic polyhedra and other (e.g., polynomial) KL invariants are the open field for research.

Another interesting question is the connection between braid family representatives (Gittings, 2004; Jablan and Sazdanovic, 2005b) and the new basic polyhedra notation introduced.

With the new notation of basic polyhedra and canonical orientation of algebraic tangles substituting their vertices, instead of the tables (or data base) of basic polyhedra similar to classic knot tables, where every KL is given by its ordering number that contains no information about it, we hope to be able to establish the complete hierarchical order in the world of polyhedral KLs in the same way as it can be established for algebraic KLs (rational, stellar, arborescent, etc.)


 

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