Basic graph theory

In the same way generating s-sequences for n=4,5,6,¼ can be derived. The s-sequence corresponding to the n-antiprismatic basic polyhedra (2n)^* (n=3,4,¼) is denoted by (0).

If we accept the (n,s,k,l)-minimality criterion, this list can be minimized by deleting the s-sequences 3 2 1 3, 3 2 1 4 3, 3 2 6 5 4, 6 1 2 1 3, 3 2 1 2 1 3, 3 2 1 3 4 3, 3 2 3 2 1 3, 3 2 4 6 5 4, 3 4 3 2 1 3, 6 1 2 6 5 4, 3 2 1 2 1 4 3, 3 2 1 2 6 5 4, 3 2 1 3 2 1 3, 3 2 1 4 5 4 3, 3 2 3 2 1 4 3, 3 2 3 2 6 5 4, 3 2 4 3 2 1 3, 3 4 3 2 1 4 3, 3 4 3 2 6 5 4, 6 1 2 1 2 1 3, 6 1 2 3 2 1 3, 3 2 1 2 1 2 1 3, 3 2 1 2 3 2 1 3, 3 2 1 4 3 2 1 3, 3 2 1 6 1 2 1 3, 6 1 2 1 3 2 1 3, and adding the s-sequences 6 1 6 5 4, 3 2 1 6 5 4, 3 2 1 2 3 2 3, 6 1 2 1 3 2 3.

In this case the minimality criterion is the minimal length of the generating sequence s and the lexicographic order. The use of (n,s,k,l)-construction instead of (n,s,k)-construction results in certain differences with regard to the preceding tables. For the generating sequences s=3 and s=6 the results are same as before, but the basic polyhedron 12D=(3,(3,2,3),3,4) will be obtained from a shorter sequence s=3 2 3 as |2| 1 2 1 3 2 3 1 2 1 2, and not from the sequence s=3 2 1 3 as |2| 1 2 1 3 2 1 3 2 1 2. In the same way, from s=3 4 3 we obtain two basic polyhedra with 12 crossings 12K=(3,(3,4,3),4,-3), 12H=(3,(3,4,3),2,-5), etc., so (n,s,k,l)-minimality criterion is more economical for the basic polyhedra notation.

By using (n,s,k,l)-minimality criterion, the tables of the basic polyhedra remain the same for n=6,8,9,10,11 crossings, but for the basic polyhedra with n=12 crossings we have new table

(3,(0),12,0) 12A (3,(3),4,-5) 12F

(3,(3),6,-3) 12B (3,(3,2,3),3,4) 12D

(3,(3,4,3),2,-5) 12H (3,(3,4,3),4,-3) 12K

(3,(3,2,4,3),3,-3) 12L (4,(3,2,3),2,4) 12I

(4,(3,2,4,3),4,1) 12G (4,(5,4,3,4,3),1,-3) 12E

(4,(3,2,1,3,2,3),3,0) 12C (4,(8,7,6,5,6,5),1,2) 12J

and for n=13 crossings the table

(3,(3),5,-5) 1312^* (3,(3),7,-3) 133^*

(3,(6),5,5) 1316^* (3,(3,2,3),4,4) 1314^*

(3,(3,2,3),5,-3) 1310^* (3,(3,4,3),4,-4) 1315^*

(3,(6,1,6),3,5) 137^* (3,(3,2,1,6),3,4) 1317^*

(3,(6,1,2,3),3,4) 138^* (4,(3,2,3),2,-5) 139^*

(4,(3,2,3),5,2) 132^* (4,(3,2,1,2,3),4,-1) 134^*

(4,(3,2,4,3,2,3),2,2) 131^* (4,(6,5,4,3,2,3),2,-2) 135^*

(4,(8,1,8,1,2,3),1,-3) 1319^* (4,(5,4,3,2,1,2,5),1,2) 1311^*

(4,(8,7,6,5,4,3,4),1,-2) 1313^* (4,(8,7,6,5,4,3,4),1,2) 136^*

(5,(3,2,4,3,4),1,-3) 1318^*

Analogous (n,s,k,l)-tables can be obtained for n=14,15,¼