$12^{2}_{182}$ - Minimal pinning sets
Pinning sets for 12^2_182 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_182 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 272
of which optimal: 3
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0386
on average over minimal pinning sets: 2.46667
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {2, 3, 7, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {2, 3, 6, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 3, 6, 7, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.4
6
0
1
19
2.68
7
0
0
55
2.89
8
0
0
81
3.04
9
0
0
69
3.15
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
1
268
Other information about this multiloop 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],[0,5,5,2],[0,1,5,6],[0,7,7,8],[0,8,6,5],[1,4,2,1],[2,4,8,9],[3,9,9,3],[3,9,6,4],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,10,14,9],[11,8,12,9],[5,19,6,20],[1,16,2,15],[10,15,11,14],[2,7,3,8],[18,4,19,5],[6,17,7,16],[3,17,4,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(11,4,-12,-5)(8,5,-9,-6)(6,17,-7,-18)(18,7,-19,-8)(19,10,-20,-11)(20,3,-13,-4)(12,13,-1,-14)(2,15,-3,-16)(9,16,-10,-17)
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)(-2,-16,9,5,-12,-14)(-3,20,10,16)(-4,11,-20)(-5,8,-19,-11)(-6,-18,-8)(-7,18)(-9,-17,6)(-10,19,7,17)(-13,12,4)(-15,2)(1,13,3,15)
Multiloop annotated with half-edges
12^2_182 annotated with half-edges