$12^{2}_{298}$ - Minimal pinning sets
Pinning sets for 12^2_298 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_298 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 448
of which optimal: 6
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.17463
on average over minimal pinning sets: 2.83333
on average over optimal pinning sets: 2.83333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 7, 8, 11}
5
[2, 3, 3, 3, 3]
2.80
B (optimal)
• {2, 4, 6, 7, 11}
5
[2, 3, 3, 3, 3]
2.80
C (optimal)
• {2, 5, 6, 7, 11}
5
[2, 3, 3, 3, 4]
3.00
D (optimal)
• {1, 3, 6, 7, 12}
5
[2, 3, 3, 3, 3]
2.80
E (optimal)
• {1, 2, 6, 7, 12}
5
[2, 3, 3, 3, 3]
2.80
F (optimal)
• {1, 2, 6, 7, 11}
5
[2, 3, 3, 3, 3]
2.80
a (minimal)
• {1, 3, 4, 7, 8, 12}
6
[2, 3, 3, 3, 3, 3]
2.83
b (minimal)
• {1, 2, 4, 7, 8, 12}
6
[2, 3, 3, 3, 3, 3]
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
2.83
6
0
2
36
2.98
7
0
0
99
3.1
8
0
0
137
3.18
9
0
0
108
3.25
10
0
0
48
3.3
11
0
0
11
3.32
12
0
0
1
3.33
Total
6
2
440
Other information about this multiloop Properties
Region degree sequence: [2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,6],[0,7,7,8],[0,9,5,1],[1,4,9,2],[2,9,8,7],[3,6,8,3],[3,7,6,9],[4,8,6,5]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,15,6,14],[3,13,4,14],[10,19,11,20],[1,16,2,15],[6,2,7,3],[17,12,18,13],[18,9,19,10],[11,9,12,8],[16,8,17,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(7,10,-8,-11)(17,8,-18,-9)(16,11,-17,-12)(1,12,-2,-13)(13,20,-14,-5)(14,3,-15,-4)(9,18,-10,-19)(2,19,-3,-20)(6,15,-7,-16)
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,-13,-5)(-2,-20,13)(-3,14,20)(-4,5,-14)(-6,-16,-12,1)(-7,-11,16)(-8,17,11)(-9,-19,2,12,-17)(-10,7,15,3,19)(-15,6,4)(-18,9)(8,10,18)
Multiloop annotated with half-edges
12^2_298 annotated with half-edges