$12^{1}_{347}$ - Minimal pinning sets
Pinning sets for 12^1_347 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_347 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 688
of which optimal: 2
of which minimal: 11
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.2016
on average over minimal pinning sets: 2.95909
on average over optimal pinning sets: 2.875
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 9, 10}
4
[2, 3, 3, 3]
2.75
B (optimal)
• {1, 4, 6, 10}
4
[2, 3, 3, 4]
3.00
a (minimal)
• {1, 2, 3, 5, 9}
5
[2, 3, 3, 3, 3]
2.80
b (minimal)
• {1, 2, 3, 5, 10}
5
[2, 3, 3, 3, 3]
2.80
c (minimal)
• {1, 4, 7, 10, 11}
5
[2, 3, 3, 4, 4]
3.20
d (minimal)
• {1, 4, 8, 9, 12}
5
[2, 3, 3, 4, 4]
3.20
e (minimal)
• {1, 4, 5, 9, 12}
5
[2, 3, 3, 3, 4]
3.00
f (minimal)
• {1, 4, 5, 7, 10}
5
[2, 3, 3, 3, 4]
3.00
g (minimal)
• {1, 3, 4, 10, 11}
5
[2, 3, 3, 3, 4]
3.00
h (minimal)
• {1, 3, 4, 8, 9}
5
[2, 3, 3, 3, 4]
3.00
i (minimal)
• {1, 3, 4, 5, 10}
5
[2, 3, 3, 3, 3]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.88
5
0
9
15
3.0
6
0
0
93
3.1
7
0
0
179
3.17
8
0
0
197
3.23
9
0
0
130
3.27
10
0
0
51
3.3
11
0
0
11
3.32
12
0
0
1
3.33
Total
2
9
677
Other information about this loop Properties
Region degree sequence: [2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,7,8],[0,8,8,4],[0,3,9,1],[1,9,7,6],[1,5,7,2],[2,6,5,9],[2,9,3,3],[4,8,7,5]]
PD code (use to draw this loop with SnapPy): [[20,15,1,16],[16,6,17,5],[12,19,13,20],[14,7,15,8],[1,7,2,6],[17,10,18,11],[11,4,12,5],[18,3,19,4],[13,9,14,8],[2,9,3,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-17,-2)(9,2,-10,-3)(3,14,-4,-15)(4,19,-5,-20)(12,5,-13,-6)(6,11,-7,-12)(20,7,-1,-8)(15,8,-16,-9)(18,13,-19,-14)(10,17,-11,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,8)(-2,9,-16)(-3,-15,-9)(-4,-20,-8,15)(-5,12,-7,20)(-6,-12)(-10,-18,-14,3)(-11,6,-13,18)(-17,10,2)(-19,4,14)(1,7,11,17)(5,19,13)
Loop annotated with half-edges
12^1_347 annotated with half-edges