$12^{2}_{72}$ - Minimal pinning sets
Pinning sets for 12^2_72 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_72 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 318
of which optimal: 8
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03847
on average over minimal pinning sets: 2.55
on average over optimal pinning sets: 2.55
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 7, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 6, 7, 9}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {1, 3, 5, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 3, 4, 7, 9}
5
[2, 2, 2, 3, 4]
2.60
E (optimal)
• {1, 2, 5, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
F (optimal)
• {1, 2, 3, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
G (optimal)
• {1, 3, 7, 9, 10}
5
[2, 2, 2, 3, 4]
2.60
H (optimal)
• {1, 3, 7, 9, 11}
5
[2, 2, 2, 3, 6]
3.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
8
0
0
2.55
6
0
0
34
2.77
7
0
0
71
2.94
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
8
0
310
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,2,2],[0,1,1,3],[0,2,5,6],[0,6,7,1],[3,7,8,8],[3,8,9,4],[4,9,9,5],[5,9,6,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[16,13,1,14],[14,8,15,7],[15,6,16,7],[12,5,13,6],[1,9,2,8],[11,20,12,17],[4,9,5,10],[2,18,3,17],[19,10,20,11],[3,18,4,19]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,2,-10,-3)(12,3,-13,-4)(4,11,-5,-12)(5,14,-6,-15)(1,6,-2,-7)(18,7,-19,-8)(13,10,-14,-11)(8,19,-9,-20)(20,15,-17,-16)(16,17,-1,-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,-7,18)(-2,9,19,7)(-3,12,-5,-15,20,-9)(-4,-12)(-6,1,17,15)(-8,-20,-16,-18)(-10,13,3)(-11,4,-13)(-14,5,11)(-17,16)(-19,8)(2,6,14,10)
Multiloop annotated with half-edges
12^2_72 annotated with half-edges