$12^{1}_{341}$ - Minimal pinning sets
Pinning sets for 12^1_341 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_341 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 372
of which optimal: 6
of which minimal: 9
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.06461
on average over minimal pinning sets: 2.6
on average over optimal pinning sets: 2.56667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 7, 11}
5
[2, 2, 2, 3, 4]
2.60
B (optimal)
• {1, 2, 4, 10, 11}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {1, 4, 7, 10, 11}
5
[2, 2, 2, 4, 4]
2.80
D (optimal)
• {1, 4, 8, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
E (optimal)
• {1, 3, 4, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
F (optimal)
• {1, 4, 8, 10, 11}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 2, 3, 4, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 4, 6, 9, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
c (minimal)
• {1, 4, 6, 7, 9, 11}
6
[2, 2, 2, 3, 4, 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
6
0
0
2.57
6
0
3
34
2.79
7
0
0
89
2.97
8
0
0
112
3.09
9
0
0
82
3.18
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
6
3
363
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,6,7],[0,7,7,8],[0,8,9,1],[1,9,9,6],[1,5,2,2],[2,8,3,3],[3,7,9,4],[4,8,5,5]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[17,4,18,5],[19,8,20,9],[6,14,7,13],[1,16,2,17],[3,10,4,11],[18,10,19,9],[7,14,8,15],[15,12,16,13],[2,12,3,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(16,5,-17,-6)(6,15,-7,-16)(20,7,-1,-8)(8,19,-9,-20)(12,9,-13,-10)(10,3,-11,-4)(4,11,-5,-12)(18,13,-19,-14)(2,17,-3,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-19,8)(-2,-18,-14)(-3,10,-13,18)(-4,-12,-10)(-5,16,-7,20,-9,12)(-6,-16)(-8,-20)(-11,4)(-15,6,-17,2)(1,7,15)(3,17,5,11)(9,19,13)
Loop annotated with half-edges
12^1_341 annotated with half-edges