$12^{1}_{52}$ - Minimal pinning sets
Pinning sets for 12^1_52 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_52 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 418
of which optimal: 3
of which minimal: 14
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.13492
on average over minimal pinning sets: 2.76837
on average over optimal pinning sets: 2.8
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 3, 3, 4]
2.80
B (optimal)
• {2, 3, 6, 7, 11}
5
[2, 2, 3, 4, 4]
3.00
C (optimal)
• {2, 3, 5, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 2, 4, 8, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 2, 4, 5, 11, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
c (minimal)
• {1, 2, 4, 5, 8, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
d (minimal)
• {1, 2, 3, 8, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
e (minimal)
• {1, 2, 3, 7, 8, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
f (minimal)
• {2, 3, 6, 9, 11, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
g (minimal)
• {2, 3, 6, 8, 9, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
h (minimal)
• {2, 3, 5, 6, 11, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
i (minimal)
• {2, 3, 4, 5, 11, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
j (minimal)
• {2, 3, 4, 5, 7, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
k (minimal)
• {1, 2, 4, 6, 9, 11, 12}
7
[2, 2, 3, 3, 3, 3, 4]
2.86
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.8
6
0
10
21
2.89
7
0
1
94
3.03
8
0
0
134
3.15
9
0
0
101
3.22
10
0
0
43
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
3
11
404
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,3],[0,2,7,8],[0,5,1,1],[1,4,6,2],[2,5,9,7],[3,6,9,8],[3,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[2,11,3,12],[14,19,15,20],[4,15,5,16],[1,13,2,12],[13,10,14,11],[18,9,19,10],[5,9,6,8],[16,8,17,7],[17,6,18,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,4,-16,-5)(5,14,-6,-15)(6,3,-7,-4)(16,7,-17,-8)(11,8,-12,-9)(20,9,-1,-10)(10,19,-11,-20)(2,13,-3,-14)(12,17,-13,-18)(1,18,-2,-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,-19,10)(-2,-14,5,-16,-8,11,19)(-3,6,14)(-4,15,-6)(-5,-15)(-7,16,4)(-9,20,-11)(-10,-20)(-12,-18,1,9)(-13,2,18)(-17,12,8)(3,13,17,7)
Loop annotated with half-edges
12^1_52 annotated with half-edges