$12^{2}_{87}$ - Minimal pinning sets
Pinning sets for 12^2_87 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_87 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 248
of which optimal: 5
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97498
on average over minimal pinning sets: 2.32
on average over optimal pinning sets: 2.32
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 7, 9}
5
[2, 2, 2, 2, 4]
2.40
B (optimal)
• {1, 3, 5, 6, 9}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 3, 4, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
D (optimal)
• {1, 3, 6, 9, 10}
5
[2, 2, 2, 2, 3]
2.20
E (optimal)
• {1, 3, 6, 8, 9}
5
[2, 2, 2, 2, 4]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
5
0
0
2.32
6
0
0
25
2.64
7
0
0
55
2.85
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
5
0
243
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: [[1,2,3,4],[0,4,4,5],[0,5,6,6],[0,6,7,7],[0,5,1,1],[1,4,8,2],[2,9,3,2],[3,9,8,3],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[3,12,4,1],[2,20,3,13],[15,11,16,12],[4,10,5,9],[1,14,2,13],[14,19,15,20],[10,16,11,17],[5,8,6,9],[6,18,7,19],[17,7,18,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,5,-19,-6)(3,6,-4,-7)(7,2,-8,-3)(16,9,-17,-10)(1,10,-2,-11)(8,17,-9,-18)(4,19,-5,-20)(15,20,-16,-13)(12,13,-1,-14)(14,11,-15,-12)
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,-11,14)(-2,7,-4,-20,15,11)(-3,-7)(-5,18,-9,16,20)(-6,3,-8,-18)(-10,1,13,-16)(-12,-14)(-13,12,-15)(-17,8,2,10)(-19,4,6)(5,19)(9,17)
Multiloop annotated with half-edges
12^2_87 annotated with half-edges