$12^{2}_{74}$ - Minimal pinning sets
Pinning sets for 12^2_74 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_74 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, 4, 5, 7, 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: [[1,1,2,3],[0,3,4,0],[0,4,4,5],[0,5,6,1],[1,7,2,2],[2,8,6,3],[3,5,8,7],[4,6,9,9],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[16,20,1,17],[17,15,18,16],[7,19,8,20],[1,14,2,15],[18,6,19,7],[8,13,9,14],[2,9,3,10],[10,5,11,6],[12,3,13,4],[4,11,5,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(17,2,-18,-3)(14,5,-15,-6)(16,7,-1,-8)(4,9,-5,-10)(13,10,-14,-11)(20,11,-17,-12)(8,15,-9,-16)(3,18,-4,-19)(12,19,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,-15,8)(-2,17,11,-14,-6)(-3,-19,12,-17)(-4,-10,13,19)(-5,14,10)(-7,16,-9,4,18,2)(-8,-16)(-11,20,-13)(-12,-20)(-18,3)(1,7)(5,9,15)
Multiloop annotated with half-edges
12^2_74 annotated with half-edges