$12^{1}_{187}$ - Minimal pinning sets
Pinning sets for 12^1_187 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_187 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 312
of which optimal: 4
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04935
on average over minimal pinning sets: 2.55
on average over optimal pinning sets: 2.45
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 6, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 3, 6, 10, 11}
5
[2, 2, 2, 3, 4]
2.60
D (optimal)
• {1, 3, 6, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 3, 4, 5, 10, 11}
6
[2, 2, 2, 3, 4, 4]
2.83
b (minimal)
• {1, 3, 4, 5, 9, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
4
0
0
2.45
6
0
2
24
2.72
7
0
0
68
2.92
8
0
0
94
3.07
9
0
0
75
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
4
2
306
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,7,7],[0,7,8,9],[1,9,2,1],[2,8,7,3],[3,6,4,3],[4,6,9,9],[4,8,8,5]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,18,9,17],[19,16,20,17],[13,6,14,7],[1,5,2,4],[18,10,19,9],[12,15,13,16],[5,14,6,15],[2,12,3,11],[3,10,4,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,18,-2,-19)(10,3,-11,-4)(17,4,-18,-5)(6,15,-7,-16)(20,7,-1,-8)(8,19,-9,-20)(2,11,-3,-12)(9,12,-10,-13)(16,13,-17,-14)(14,5,-15,-6)
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,-19,8)(-2,-12,9,19)(-3,10,12)(-4,17,13,-10)(-5,14,-17)(-6,-16,-14)(-7,20,-9,-13,16)(-8,-20)(-11,2,18,4)(-15,6)(-18,1,7,15,5)(3,11)
Loop annotated with half-edges
12^1_187 annotated with half-edges