$12^{2}_{107}$ - Minimal pinning sets
Pinning sets for 12^2_107 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_107 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 140
of which optimal: 3
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98196
on average over minimal pinning sets: 2.43254
on average over optimal pinning sets: 2.38889
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 6, 9, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 3, 4, 6, 9, 10}
6
[2, 2, 2, 2, 3, 4]
2.50
C (optimal)
• {1, 2, 3, 6, 9, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
a (minimal)
• {1, 3, 5, 6, 8, 9, 11}
7
[2, 2, 2, 2, 3, 3, 3]
2.43
b (minimal)
• {1, 3, 4, 6, 8, 9, 11}
7
[2, 2, 2, 2, 3, 3, 4]
2.57
c (minimal)
• {1, 2, 3, 6, 8, 9, 11}
7
[2, 2, 2, 2, 3, 3, 3]
2.43
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
3
0
0
2.39
7
0
3
15
2.65
8
0
0
40
2.89
9
0
0
44
3.06
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
3
134
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],[0,3,4,4],[0,5,6,0],[0,6,6,1],[1,7,8,1],[2,9,7,6],[2,5,3,3],[4,5,9,8],[4,7,9,9],[5,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,12,6,13],[19,3,20,4],[1,11,2,12],[6,14,7,13],[9,18,10,19],[10,2,11,3],[14,18,15,17],[7,17,8,16],[8,15,9,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(14,1,-15,-2)(3,6,-4,-7)(16,7,-17,-8)(8,17,-9,-18)(19,10,-20,-11)(11,18,-12,-19)(12,9,-13,-10)(20,13,-5,-14)(2,15,-3,-16)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-5)(-2,-16,-8,-18,11,-20,-14)(-3,-7,16)(-4,5,13,9,17,7)(-6,3,15,1)(-9,12,18)(-10,19,-12)(-11,-19)(-13,20,10)(-15,2)(-17,8)(4,6)
Multiloop annotated with half-edges
12^2_107 annotated with half-edges