$12^{2}_{184}$ - Minimal pinning sets
Pinning sets for 12^2_184 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_184 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, 3, 6, 10}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 5, 6, 8}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 2, 3, 6, 8, 9}
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, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,7],[1,8,8,5],[2,4,9,6],[3,5,9,9],[3,9,8,8],[4,7,7,4],[5,7,6,6]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,7,10,8],[10,19,11,20],[1,11,2,12],[14,6,15,7],[15,18,16,19],[2,16,3,17],[12,4,13,5],[5,13,6,14],[17,3,18,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(17,2,-18,-3)(13,6,-14,-7)(20,7,-9,-8)(8,9,-1,-10)(5,12,-6,-13)(11,14,-12,-15)(18,15,-19,-16)(3,16,-4,-17)(4,19,-5,-20)
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)(-2,17,-4,-20,-8,-10)(-3,-17)(-5,-13,-7,20)(-6,13)(-9,8)(-11,-15,18,2)(-12,5,19,15)(-14,11,1,9,7)(-16,3,-18)(-19,4,16)(6,12,14)
Multiloop annotated with half-edges
12^2_184 annotated with half-edges