$12^{1}_{114}$ - Minimal pinning sets
Pinning sets for 12^1_114 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_114 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 324
of which optimal: 4
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05128
on average over minimal pinning sets: 2.54167
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, 11, 12}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 4, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 3, 4, 7, 11}
5
[2, 2, 2, 3, 4]
2.60
D (optimal)
• {1, 2, 3, 7, 11}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 2, 3, 9, 11, 12}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 3, 8, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 2, 3, 5, 11, 12}
6
[2, 2, 2, 3, 3, 4]
2.67
d (minimal)
• {1, 2, 3, 5, 8, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
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
2.5
6
0
4
24
2.73
7
0
0
73
2.93
8
0
0
98
3.07
9
0
0
76
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
4
4
316
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,3],[0,2,7,7],[0,5,1,1],[1,4,8,2],[2,8,9,9],[3,9,8,3],[5,7,9,6],[6,8,7,6]]
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],[7,18,8,19],[5,17,6,16],[6,9,7,10],[17,8,18,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,6,-16,-7)(4,7,-5,-8)(11,8,-12,-9)(20,9,-1,-10)(10,19,-11,-20)(2,13,-3,-14)(14,3,-15,-4)(5,16,-6,-17)(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,-4,-8,11,19)(-3,14)(-5,-17,12,8)(-6,15,3,13,17)(-7,4,-15)(-9,20,-11)(-10,-20)(-12,-18,1,9)(-13,2,18)(-16,5,7)(6,16)
Loop annotated with half-edges
12^1_114 annotated with half-edges