$12^{2}_{122}$ - Minimal pinning sets
Pinning sets for 12^2_122 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_122 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, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 4, 6, 9}
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, 3, 3, 4, 6, 7]
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],[1,3,7,8],[2,9,9,6],[2,5,9,7],[4,6,8,8],[4,7,7,9],[5,8,6,5]]
PD code (use to draw this multiloop with SnapPy): [[3,14,4,1],[2,20,3,15],[13,4,14,5],[1,16,2,15],[16,19,17,20],[5,11,6,10],[12,9,13,10],[18,8,19,9],[17,8,18,7],[11,7,12,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,2,-12,-3)(4,9,-5,-10)(5,18,-6,-19)(19,6,-20,-7)(10,7,-11,-8)(8,3,-9,-4)(1,12,-2,-13)(17,20,-18,-15)(14,15,-1,-16)(16,13,-17,-14)
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,-13,16)(-2,11,7,-20,17,13)(-3,8,-11)(-4,-10,-8)(-5,-19,-7,10)(-6,19)(-9,4)(-12,1,15,-18,5,9,3)(-14,-16)(-15,14,-17)(2,12)(6,18,20)
Multiloop annotated with half-edges
12^2_122 annotated with half-edges