$12^{2}_{15}$ - Minimal pinning sets
Pinning sets for 12^2_15 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_15 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 919
of which optimal: 13
of which minimal: 13
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.14021
on average over minimal pinning sets: 2.76923
on average over optimal pinning sets: 2.76923
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {2, 3, 5, 12}
4
[2, 2, 3, 4]
2.75
C (optimal)
• {2, 4, 5, 11}
4
[2, 2, 4, 4]
3.00
D (optimal)
• {1, 2, 5, 9}
4
[2, 2, 3, 4]
2.75
E (optimal)
• {2, 5, 9, 12}
4
[2, 2, 4, 4]
3.00
F (optimal)
• {2, 4, 5, 7}
4
[2, 2, 4, 4]
3.00
G (optimal)
• {1, 2, 5, 6}
4
[2, 2, 3, 3]
2.50
H (optimal)
• {2, 5, 6, 12}
4
[2, 2, 3, 4]
2.75
I (optimal)
• {2, 3, 5, 8}
4
[2, 2, 3, 3]
2.50
J (optimal)
• {2, 5, 8, 9}
4
[2, 2, 3, 4]
2.75
K (optimal)
• {2, 5, 6, 8}
4
[2, 2, 3, 3]
2.50
L (optimal)
• {2, 5, 10, 11}
4
[2, 2, 4, 4]
3.00
M (optimal)
• {2, 5, 7, 10}
4
[2, 2, 4, 4]
3.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
13
0
0
2.77
5
0
0
82
2.96
6
0
0
190
3.07
7
0
0
248
3.14
8
0
0
210
3.2
9
0
0
120
3.24
10
0
0
45
3.28
11
0
0
10
3.31
12
0
0
1
3.33
Total
13
0
906
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,7],[0,7,6,5],[0,5,1,1],[1,4,3,2],[2,3,8,9],[2,9,8,3],[6,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[3,10,4,1],[2,20,3,11],[13,9,14,10],[4,18,5,19],[1,12,2,11],[12,19,13,20],[5,8,6,9],[14,17,15,18],[15,7,16,8],[6,16,7,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,5,-17,-6)(3,6,-4,-7)(14,7,-15,-8)(1,8,-2,-9)(4,17,-5,-18)(15,18,-16,-19)(2,19,-3,-20)(13,20,-14,-11)(10,11,-1,-12)(12,9,-13,-10)
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,-9,12)(-2,-20,13,9)(-3,-7,14,20)(-4,-18,15,7)(-5,16,18)(-6,3,19,-16)(-8,1,11,-14)(-10,-12)(-11,10,-13)(-15,-19,2,8)(-17,4,6)(5,17)
Multiloop annotated with half-edges
12^2_15 annotated with half-edges