$12^{2}_{144}$ - Minimal pinning sets
Pinning sets for 12^2_144 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_144 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, 5, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 5, 8, 11}
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: [[1,2,3,3],[0,3,4,5],[0,6,7,8],[0,8,1,0],[1,7,6,5],[1,4,6,6],[2,5,5,4],[2,4,9,9],[2,9,9,3],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,19,8,18],[5,15,6,16],[19,1,20,2],[8,12,9,11],[17,10,18,11],[16,10,17,9],[12,4,13,5],[14,2,15,3],[3,13,4,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(19,2,-20,-3)(12,5,-13,-6)(16,9,-17,-10)(10,17,-11,-18)(18,15,-19,-16)(1,20,-2,-7)(3,8,-4,-9)(11,14,-12,-15)(4,13,-5,-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,-7)(-2,19,15,-12,-6,7)(-3,-9,16,-19)(-4,-14,11,17,9)(-5,12,14)(-8,3,-20,1)(-10,-18,-16)(-11,-15,18)(-13,4,8,6)(-17,10)(2,20)(5,13)
Multiloop annotated with half-edges
12^2_144 annotated with half-edges