$12^{2}_{142}$ - Minimal pinning sets
Pinning sets for 12^2_142 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_142 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 176
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97631
on average over minimal pinning sets: 2.34444
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 4, 5, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 2, 4, 5, 9, 12}
6
[2, 2, 2, 2, 3, 4]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
2
7
2.5
7
0
0
30
2.75
8
0
0
51
2.94
9
0
0
49
3.08
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
2
173
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,5,6,6],[0,7,8,4],[1,3,5,1],[2,4,9,6],[2,5,7,2],[3,6,9,8],[3,7,9,9],[5,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,9,12,10],[5,19,6,20],[1,14,2,13],[8,12,9,13],[4,7,5,8],[18,6,19,7],[14,18,15,17],[2,17,3,16],[3,15,4,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,4,-14,-5)(2,5,-3,-6)(16,7,-17,-8)(18,9,-19,-10)(1,20,-2,-11)(11,10,-12,-1)(12,19,-13,-20)(3,14,-4,-15)(8,15,-9,-16)(6,17,-7,-18)
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,-6,-18,-10,11)(-3,-15,8,-17,6)(-4,13,19,9,15)(-5,2,20,-13)(-7,16,-9,18)(-8,-16)(-12,-20,1)(-14,3,5)(-19,12,10)(4,14)(7,17)
Multiloop annotated with half-edges
12^2_142 annotated with half-edges