$12^{1}_{186}$ - Minimal pinning sets
Pinning sets for 12^1_186 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_186 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, 3, 4, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 4, 6, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 4, 8, 11, 12}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 2, 4, 8, 11}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 4, 6, 7, 11, 12}
6
[2, 2, 2, 3, 3, 4]
2.67
b (minimal)
• {1, 2, 4, 6, 7, 11}
6
[2, 2, 2, 3, 4, 4]
2.83
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],[0,8,8,9],[1,9,2,1],[2,9,7,3],[3,6,8,3],[4,7,9,4],[4,8,6,5]]
PD code (use to draw this loop with SnapPy): [[20,5,1,6],[6,18,7,17],[19,16,20,17],[13,4,14,5],[1,10,2,11],[18,8,19,7],[12,15,13,16],[3,14,4,15],[9,2,10,3],[11,9,12,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(11,2,-12,-3)(18,3,-19,-4)(5,16,-6,-17)(6,9,-7,-10)(19,8,-20,-9)(1,12,-2,-13)(10,13,-11,-14)(17,14,-18,-15)(15,4,-16,-5)
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,-13,10,-7)(-2,11,13)(-3,18,14,-11)(-4,15,-18)(-5,-17,-15)(-6,-10,-14,17)(-8,19,3,-12,1)(-9,6,16,4,-19)(-16,5)(-20,7,9)(2,12)(8,20)
Loop annotated with half-edges
12^1_186 annotated with half-edges