$12^{2}_{181}$ - Minimal pinning sets
Pinning sets for 12^2_181 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_181 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 360
of which optimal: 8
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04549
on average over minimal pinning sets: 2.5
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, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 3, 4, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
D (optimal)
• {1, 2, 3, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
E (optimal)
• {1, 3, 4, 9, 12}
5
[2, 2, 2, 3, 3]
2.40
F (optimal)
• {1, 2, 3, 9, 12}
5
[2, 2, 2, 3, 3]
2.40
G (optimal)
• {1, 3, 4, 8, 9}
5
[2, 2, 2, 3, 4]
2.60
H (optimal)
• {1, 2, 3, 8, 9}
5
[2, 2, 2, 3, 4]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
8
0
0
2.5
6
0
0
40
2.77
7
0
0
86
2.96
8
0
0
104
3.08
9
0
0
77
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
8
0
352
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],[0,5,5,2],[0,1,5,6],[0,7,7,8],[0,8,9,5],[1,4,2,1],[2,9,9,7],[3,6,8,3],[3,7,9,4],[4,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,6,10,5],[7,4,8,5],[14,19,15,20],[1,12,2,11],[6,11,7,10],[17,3,18,4],[18,13,19,14],[15,13,16,12],[2,16,3,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(5,2,-6,-3)(3,14,-4,-15)(15,4,-16,-5)(16,7,-17,-8)(17,20,-18,-9)(11,18,-12,-19)(8,9,-1,-10)(19,12,-20,-13)(6,13,-7,-14)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,5,-16,-8,-10)(-3,-15,-5)(-4,15)(-6,-14,3)(-7,16,4,14)(-9,8,-17)(-11,-19,-13,6,2)(-12,19)(-18,11,1,9)(-20,17,7,13)(12,18,20)
Multiloop annotated with half-edges
12^2_181 annotated with half-edges