$12^{1}_{68}$ - Minimal pinning sets
Pinning sets for 12^1_68 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_68 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 304
of which optimal: 5
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04294
on average over minimal pinning sets: 2.48
on average over optimal pinning sets: 2.48
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 5, 6, 7}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 5, 7, 9}
5
[2, 2, 2, 3, 4]
2.60
D (optimal)
• {1, 2, 5, 7, 8}
5
[2, 2, 2, 3, 4]
2.60
E (optimal)
• {2, 3, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
5
0
0
2.48
6
0
0
28
2.74
7
0
0
67
2.93
8
0
0
89
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
5
0
299
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,6,7],[0,7,8,4],[1,3,5,5],[1,4,4,2],[2,8,9,9],[2,9,8,3],[3,7,9,6],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[20,15,1,16],[16,19,17,20],[14,7,15,8],[1,11,2,10],[18,9,19,10],[17,9,18,8],[13,4,14,5],[6,11,7,12],[2,6,3,5],[3,12,4,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,2,-12,-3)(3,20,-4,-1)(12,5,-13,-6)(6,15,-7,-16)(16,7,-17,-8)(8,19,-9,-20)(4,9,-5,-10)(1,10,-2,-11)(18,13,-19,-14)(14,17,-15,-18)
Edge permutation $\epsilon=$ (-1,1)(-2,2)(-3,3)(-4,4)(-5,5)(-6,6)(-7,7)(-8,8)(-9,9)(-10,10)(-11,11)(-12,12)(-13,13)(-14,14)(-15,15)(-16,16)(-17,17)(-18,18)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-11,-3)(-2,11)(-4,-10,1)(-5,12,2,10)(-6,-16,-8,-20,3,-12)(-7,16)(-9,4,20)(-13,18,-15,6)(-14,-18)(-17,14,-19,8)(5,9,19,13)(7,15,17)
Loop annotated with half-edges
12^1_68 annotated with half-edges