$12^{2}_{186}$ - Minimal pinning sets
Pinning sets for 12^2_186 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_186 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 256
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96564
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 9}
4
[2, 2, 2, 2]
2.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.0
5
0
0
8
2.4
6
0
0
28
2.67
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
0
255
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,5],[0,5,6,4],[1,3,7,8],[2,8,3,2],[3,9,7,7],[4,6,6,9],[4,9,9,5],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,5,8,6],[8,19,9,20],[1,18,2,17],[4,16,5,17],[18,9,19,10],[2,12,3,13],[13,3,14,4],[15,10,16,11],[11,14,12,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(15,2,-16,-3)(12,17,-13,-18)(9,18,-10,-19)(5,20,-6,-7)(19,8,-20,-9)(1,10,-2,-11)(11,4,-12,-5)(16,13,-17,-14)(3,14,-4,-15)
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,-5,-7)(-2,15,-4,11)(-3,-15)(-6,7)(-8,19,-10,1)(-9,-19)(-12,-18,9,-20,5)(-13,16,2,10,18)(-14,3,-16)(-17,12,4,14)(6,20,8)(13,17)
Multiloop annotated with half-edges
12^2_186 annotated with half-edges