$12^{1}_{343}$ - Minimal pinning sets
Pinning sets for 12^1_343 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_343 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 634
of which optimal: 4
of which minimal: 12
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.13118
on average over minimal pinning sets: 2.78889
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, 3, 8, 9}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 3, 8, 10}
4
[2, 2, 3, 4]
2.75
C (optimal)
• {1, 3, 7, 8}
4
[2, 2, 3, 4]
2.75
D (optimal)
• {1, 3, 8, 12}
4
[2, 2, 3, 5]
3.00
a (minimal)
• {1, 2, 3, 5, 8}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {1, 2, 3, 6, 10}
5
[2, 2, 3, 3, 4]
2.80
c (minimal)
• {1, 3, 6, 7, 10}
5
[2, 2, 3, 4, 4]
3.00
d (minimal)
• {1, 3, 4, 5, 8}
5
[2, 2, 3, 4, 4]
3.00
e (minimal)
• {1, 2, 3, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
f (minimal)
• {1, 3, 6, 7, 11}
5
[2, 2, 3, 3, 4]
2.80
g (minimal)
• {1, 3, 4, 8, 11}
5
[2, 2, 3, 3, 4]
2.80
h (minimal)
• {1, 2, 3, 6, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
4
0
0
2.75
5
0
7
26
2.89
6
0
1
98
3.0
7
0
0
164
3.1
8
0
0
169
3.18
9
0
0
110
3.23
10
0
0
44
3.28
11
0
0
10
3.31
12
0
0
1
3.33
Total
4
8
622
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],[0,4,5,2],[0,1,6,7],[0,7,7,8],[0,8,9,1],[1,9,6,6],[2,5,5,9],[2,8,3,3],[3,7,9,4],[4,8,6,5]]
PD code (use to draw this loop with SnapPy): [[20,15,1,16],[16,11,17,12],[12,19,13,20],[14,7,15,8],[1,10,2,11],[17,5,18,4],[18,3,19,4],[13,9,14,8],[9,6,10,7],[2,6,3,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,20,-16,-1)(1,14,-2,-15)(2,19,-3,-20)(6,3,-7,-4)(11,4,-12,-5)(5,10,-6,-11)(18,7,-19,-8)(13,8,-14,-9)(17,12,-18,-13)(9,16,-10,-17)
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,-15)(-2,-20,15)(-3,6,10,16,20)(-4,11,-6)(-5,-11)(-7,18,12,4)(-8,13,-18)(-9,-17,-13)(-10,5,-12,17)(-14,1,-16,9)(-19,2,14,8)(3,19,7)
Loop annotated with half-edges
12^1_343 annotated with half-edges