$12^{1}_{96}$ - Minimal pinning sets
Pinning sets for 12^1_96 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_96 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, 3, 5, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 5, 7, 9}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {1, 3, 5, 7, 11}
5
[2, 2, 2, 3, 4]
2.60
D (optimal)
• {1, 3, 4, 5, 7}
5
[2, 2, 2, 3, 3]
2.40
E (optimal)
• {1, 2, 4, 5, 7}
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, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,6],[0,6,7,8],[0,5,1,1],[1,4,9,2],[2,7,3,2],[3,6,9,8],[3,7,9,9],[5,8,8,7]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[13,2,14,3],[19,10,20,11],[4,18,5,17],[1,12,2,13],[14,12,15,11],[9,18,10,19],[5,9,6,8],[16,7,17,8],[15,7,16,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (2,11,-3,-12)(12,3,-13,-4)(4,1,-5,-2)(15,6,-16,-7)(7,10,-8,-11)(17,8,-18,-9)(13,20,-14,-1)(5,14,-6,-15)(19,16,-20,-17)(9,18,-10,-19)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,-13)(-2,-12,-4)(-3,12)(-5,-15,-7,-11,2)(-6,15)(-8,17,-20,13,3,11)(-9,-19,-17)(-10,7,-16,19)(-14,5,1)(-18,9)(6,14,20,16)(8,10,18)
Loop annotated with half-edges
12^1_96 annotated with half-edges