$12^{1}_{185}$ - Minimal pinning sets
Pinning sets for 12^1_185 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_185 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 252
of which optimal: 6
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97224
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 9, 10}
5
[2, 2, 2, 2, 4]
2.40
B (optimal)
• {1, 3, 4, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
C (optimal)
• {1, 3, 6, 9, 11}
5
[2, 2, 2, 2, 4]
2.40
D (optimal)
• {1, 3, 5, 6, 9}
5
[2, 2, 2, 2, 4]
2.40
E (optimal)
• {1, 2, 3, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
F (optimal)
• {1, 3, 6, 7, 9}
5
[2, 2, 2, 2, 4]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
6
0
0
2.33
6
0
0
27
2.65
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
6
0
246
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,5,6,0],[0,7,8,8],[1,8,5,1],[1,4,9,2],[2,9,7,7],[3,6,6,9],[3,9,4,3],[5,8,7,6]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[9,4,10,5],[19,6,20,7],[1,17,2,16],[3,8,4,9],[10,8,11,7],[18,13,19,14],[17,13,18,12],[2,15,3,16],[11,15,12,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(16,3,-17,-4)(4,7,-5,-8)(20,5,-1,-6)(9,14,-10,-15)(15,10,-16,-11)(11,8,-12,-9)(19,12,-20,-13)(13,18,-14,-19)(2,17,-3,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6)(-2,-18,13,-20,-6)(-3,16,10,14,18)(-4,-8,11,-16)(-5,20,12,8)(-7,4,-17,2)(-9,-15,-11)(-10,15)(-12,19,-14,9)(-13,-19)(1,5,7)(3,17)
Loop annotated with half-edges
12^1_185 annotated with half-edges