$12^{1}_{324}$ - Minimal pinning sets
Pinning sets for 12^1_324 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_324 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 450
of which optimal: 8
of which minimal: 10
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.12673
on average over minimal pinning sets: 2.76667
on average over optimal pinning sets: 2.75
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 12}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 4, 5, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 2, 5, 7, 12}
5
[2, 2, 3, 3, 4]
2.80
D (optimal)
• {1, 3, 5, 7, 12}
5
[2, 2, 3, 4, 4]
3.00
E (optimal)
• {1, 2, 5, 8, 12}
5
[2, 2, 3, 3, 3]
2.60
F (optimal)
• {1, 3, 5, 8, 12}
5
[2, 2, 3, 3, 4]
2.80
G (optimal)
• {1, 4, 5, 8, 12}
5
[2, 2, 3, 3, 3]
2.60
H (optimal)
• {1, 4, 5, 10, 12}
5
[2, 2, 3, 3, 5]
3.00
a (minimal)
• {1, 3, 4, 5, 9, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
b (minimal)
• {1, 4, 5, 7, 11, 12}
6
[2, 2, 3, 3, 3, 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
8
0
0
2.75
6
0
2
46
2.92
7
0
0
110
3.06
8
0
0
134
3.15
9
0
0
97
3.22
10
0
0
42
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
8
2
440
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,3,3],[0,2,2,7],[0,8,5,1],[1,4,8,6],[1,5,9,2],[3,9,9,8],[4,7,9,5],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[17,20,18,1],[16,9,17,10],[4,19,5,20],[18,5,19,6],[1,11,2,10],[2,15,3,16],[3,8,4,9],[6,13,7,14],[11,14,12,15],[12,7,13,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,3,-17,-4)(10,5,-11,-6)(1,6,-2,-7)(14,7,-15,-8)(8,19,-9,-20)(4,11,-5,-12)(9,12,-10,-13)(20,13,-1,-14)(2,17,-3,-18)(15,18,-16,-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,-7,14)(-2,-18,15,7)(-3,16,18)(-4,-12,9,19,-16)(-5,10,12)(-6,1,13,-10)(-8,-20,-14)(-9,-13,20)(-11,4,-17,2,6)(-15,-19,8)(3,17)(5,11)
Loop annotated with half-edges
12^1_324 annotated with half-edges