$12^{3}_{30}$ - Minimal pinning sets
Pinning sets for 12^3_30
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_30
Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96906
on average over minimal pinning sets: 2.2
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, 7, 12}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
•
{2, 3, 4, 7, 12}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
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],[0,4,5,0],[0,5,5,6],[0,7,7,4],[1,3,7,8],[1,9,2,2],[2,9,9,8],[3,8,4,3],[4,7,6,9],[5,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[12,16,1,13],[13,11,14,12],[15,20,16,17],[1,9,2,8],[10,7,11,8],[14,18,15,17],[5,19,6,20],[9,3,10,2],[3,6,4,7],[18,4,19,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,7,-1,-8)(13,2,-14,-3)(1,4,-2,-5)(8,5,-9,-6)(6,11,-7,-12)(18,9,-19,-10)(10,17,-11,-18)(3,14,-4,-15)(20,15,-17,-16)(16,19,-13,-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,-5,8)(-2,13,19,9,5)(-3,-15,20,-13)(-4,1,7,11,17,15)(-6,-12,-8)(-7,12)(-9,18,-11,6)(-10,-18)(-14,3)(-16,-20)(-17,10,-19,16)(2,4,14)
Multiloop annotated with half-edges
12^3_30 annotated with half-edges