$12^{1}_{180}$ - Minimal pinning sets
Pinning sets for 12^1_180 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_180 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 508
of which optimal: 1
of which minimal: 9
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.11724
on average over minimal pinning sets: 2.74444
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 5}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 2, 4, 6, 9}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 2, 4, 5, 9}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {1, 2, 4, 5, 7}
5
[2, 2, 3, 3, 4]
2.80
d (minimal)
• {1, 2, 4, 5, 11}
5
[2, 2, 3, 3, 4]
2.80
e (minimal)
• {1, 3, 4, 6, 10}
5
[2, 2, 3, 3, 4]
2.80
f (minimal)
• {1, 3, 4, 6, 9}
5
[2, 2, 3, 3, 3]
2.60
g (minimal)
• {1, 2, 4, 6, 7, 10}
6
[2, 2, 3, 3, 4, 4]
3.00
h (minimal)
• {1, 2, 4, 6, 10, 11}
6
[2, 2, 3, 3, 4, 4]
3.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.5
5
0
6
8
2.73
6
0
2
59
2.92
7
0
0
127
3.06
8
0
0
148
3.16
9
0
0
103
3.23
10
0
0
43
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
8
499
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 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],[0,7,8,9],[0,9,6,5],[1,4,2,1],[2,4,9,7],[3,6,8,8],[3,7,7,9],[3,8,6,4]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[8,17,9,18],[19,16,20,17],[10,13,11,14],[1,6,2,7],[18,7,19,8],[2,15,3,16],[3,12,4,13],[11,4,12,5],[14,5,15,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(11,2,-12,-3)(18,3,-19,-4)(15,4,-16,-5)(20,9,-1,-10)(7,10,-8,-11)(19,12,-20,-13)(16,13,-17,-14)(5,14,-6,-15)(6,17,-7,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,10)(-2,11,-8)(-3,18,-7,-11)(-4,15,-6,-18)(-5,-15)(-9,20,12,2)(-10,7,17,13,-20)(-12,19,3)(-13,16,4,-19)(-14,5,-16)(-17,6,14)(1,9)
Loop annotated with half-edges
12^1_180 annotated with half-edges