$12^{2}_{327}$ - Minimal pinning sets
Pinning sets for 12^2_327 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_327 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 408
of which optimal: 10
of which minimal: 10
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.06254
on average over minimal pinning sets: 2.6
on average over optimal pinning sets: 2.6
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 7, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 4, 7, 8, 9}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {1, 4, 7, 9, 12}
5
[2, 2, 2, 4, 4]
2.80
D (optimal)
• {1, 4, 7, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
E (optimal)
• {1, 3, 4, 7, 8}
5
[2, 2, 2, 3, 4]
2.60
F (optimal)
• {1, 2, 4, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
G (optimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
H (optimal)
• {1, 4, 5, 7, 9}
5
[2, 2, 2, 4, 4]
2.80
I (optimal)
• {1, 4, 6, 7, 11}
5
[2, 2, 2, 3, 4]
2.60
J (optimal)
• {1, 4, 6, 7, 9}
5
[2, 2, 2, 4, 4]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
10
0
0
2.6
6
0
0
49
2.83
7
0
0
102
2.99
8
0
0
118
3.1
9
0
0
83
3.18
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
10
0
398
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,7,3],[0,2,8,9],[0,9,9,1],[1,9,8,8],[1,7,7,2],[2,6,6,8],[3,7,5,5],[3,5,4,4]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,15,10,14],[7,4,8,5],[19,3,20,4],[1,16,2,15],[10,17,11,18],[13,5,14,6],[6,12,7,13],[18,11,19,12],[2,16,3,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(5,2,-6,-3)(18,3,-19,-4)(11,16,-12,-17)(4,17,-5,-18)(19,14,-20,-15)(20,7,-9,-8)(8,9,-1,-10)(15,10,-16,-11)(6,13,-7,-14)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,16,10)(-2,5,17,-12)(-3,18,-5)(-4,-18)(-6,-14,19,3)(-7,20,14)(-8,-10,15,-20)(-9,8)(-11,-17,4,-19,-15)(-13,6,2)(-16,11)(1,9,7,13)
Multiloop annotated with half-edges
12^2_327 annotated with half-edges