$12^{2}_{22}$ - Minimal pinning sets
Pinning sets for 12^2_22 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_22 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, 7, 11}
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, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,3,4],[0,5,6,0],[0,4,1,1],[1,3,6,7],[2,7,7,8],[2,8,9,4],[4,9,5,5],[5,9,9,6],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[3,16,4,1],[9,2,10,3],[15,4,16,5],[1,8,2,9],[10,8,11,7],[5,17,6,20],[14,11,15,12],[6,17,7,18],[19,12,20,13],[13,18,14,19]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(4,7,-5,-8)(14,5,-15,-6)(11,8,-12,-9)(20,9,-17,-10)(16,13,-1,-14)(6,15,-7,-16)(17,2,-18,-3)(3,18,-4,-19)(10,19,-11,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,8,-5,14)(-2,17,9,-12)(-3,-19,10,-17)(-4,-8,11,19)(-6,-16,-14)(-7,4,18,2,-13,16)(-9,20,-11)(-10,-20)(-15,6)(-18,3)(1,13)(5,7,15)
Multiloop annotated with half-edges
12^2_22 annotated with half-edges