$12^{2}_{195}$ - Minimal pinning sets
Pinning sets for 12^2_195 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_195 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 136
of which optimal: 3
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97614
on average over minimal pinning sets: 2.39286
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 3, 5, 9, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 2, 3, 4, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
a (minimal)
• {1, 2, 3, 4, 7, 9, 10}
7
[2, 2, 2, 2, 3, 3, 4]
2.57
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
2.33
7
0
1
16
2.64
8
0
0
38
2.88
9
0
0
43
3.05
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
1
132
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,7,8],[0,9,9,5],[1,4,2,1],[2,7,7,3],[3,6,6,8],[3,7,9,9],[4,8,8,4]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,8,12,7],[9,6,10,7],[19,16,20,17],[1,14,2,13],[8,13,9,12],[5,17,6,18],[18,4,19,5],[15,3,16,4],[14,3,15,2]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,11,-1,-12)(12,1,-13,-2)(15,2,-16,-3)(7,4,-8,-5)(5,18,-6,-19)(19,6,-20,-7)(20,9,-11,-10)(16,13,-17,-14)(3,14,-4,-15)(8,17,-9,-18)
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)(-2,15,-4,7,-20,-10,-12)(-3,-15)(-5,-19,-7)(-6,19)(-8,-18,5)(-9,20,6,18)(-11,10)(-13,16,2)(-14,3,-16)(-17,8,4,14)(1,11,9,17,13)
Multiloop annotated with half-edges
12^2_195 annotated with half-edges