$12^{2}_{220}$ - Minimal pinning sets
Pinning sets for 12^2_220 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_220 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 144
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97193
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 6, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 3, 4, 5, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 2, 3, 5, 6, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
D (optimal)
• {1, 2, 3, 4, 5, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
4
0
0
2.33
7
0
0
20
2.66
8
0
0
41
2.89
9
0
0
44
3.06
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
4
0
140
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,4,5],[0,6,7,0],[1,7,2,1],[2,8,8,6],[3,5,9,7],[3,6,9,4],[5,9,9,5],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,8,12,7],[9,6,10,7],[19,1,20,2],[8,13,9,12],[5,15,6,16],[2,15,3,14],[18,13,19,14],[16,4,17,5],[3,17,4,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,10,-12,-1)(5,2,-6,-3)(3,16,-4,-17)(14,7,-15,-8)(17,4,-18,-5)(13,18,-14,-19)(19,8,-20,-9)(1,20,-2,-11)(9,12,-10,-13)(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,-11)(-2,5,-18,13,-10,11)(-3,-17,-5)(-4,17)(-6,-16,3)(-7,14,18,4,16)(-8,19,-14)(-9,-13,-19)(-12,9,-20,1)(-15,6,2,20,8)(7,15)(10,12)
Multiloop annotated with half-edges
12^2_220 annotated with half-edges