$12^{2}_{322}$ - Minimal pinning sets
Pinning sets for 12^2_322 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_322 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 332
of which optimal: 5
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05102
on average over minimal pinning sets: 2.6125
on average over optimal pinning sets: 2.48
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 7, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 3, 5, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 3, 4, 7, 9}
5
[2, 2, 2, 3, 4]
2.60
E (optimal)
• {1, 2, 3, 7, 9}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 3, 7, 8, 10, 11}
6
[2, 2, 2, 3, 4, 5]
3.00
b (minimal)
• {1, 3, 5, 7, 8, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
c (minimal)
• {1, 3, 4, 7, 8, 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
5
0
0
2.48
6
0
3
28
2.75
7
0
0
76
2.94
8
0
0
99
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
5
3
324
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,7],[0,7,7,3],[0,2,8,4],[0,3,5,5],[1,4,4,9],[1,9,8,7],[1,6,2,2],[3,6,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,3,8,4],[13,5,14,6],[14,19,15,20],[1,15,2,16],[16,2,17,3],[8,11,9,12],[4,12,5,13],[9,18,10,19],[17,10,18,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(12,1,-13,-2)(18,3,-19,-4)(8,15,-9,-16)(13,16,-14,-17)(2,17,-3,-18)(5,20,-6,-7)(14,9,-15,-10)(19,10,-20,-11)(4,11,-5,-12)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-5,-7)(-2,-18,-4,-12)(-3,18)(-6,7)(-8,-16,13,1)(-9,14,16)(-10,19,3,17,-14)(-11,4,-19)(-13,-17,2)(-15,8,6,20,10)(-20,5,11)(9,15)
Multiloop annotated with half-edges
12^2_322 annotated with half-edges