$12^{4}_{1}$ - Minimal pinning sets
Pinning sets for 12^4_1 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^4_1 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 196
of which optimal: 9
of which minimal: 9
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9903
on average over minimal pinning sets: 2.44444
on average over optimal pinning sets: 2.44444
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 6, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 4, 6, 9, 12}
6
[2, 2, 2, 2, 3, 4]
2.50
C (optimal)
• {1, 2, 4, 6, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
D (optimal)
• {1, 2, 3, 6, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
E (optimal)
• {1, 2, 3, 6, 9, 12}
6
[2, 2, 2, 2, 4, 4]
2.67
F (optimal)
• {1, 2, 3, 6, 8, 9}
6
[2, 2, 2, 2, 3, 4]
2.50
G (optimal)
• {1, 2, 5, 6, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
H (optimal)
• {1, 2, 5, 6, 9, 12}
6
[2, 2, 2, 2, 3, 4]
2.50
I (optimal)
• {1, 2, 5, 6, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
9
0
0
2.44
7
0
0
36
2.76
8
0
0
60
2.97
9
0
0
54
3.1
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
9
0
187
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,7,0],[0,7,7,1],[1,8,8,5],[1,4,9,9],[2,9,8,7],[2,6,3,3],[4,6,9,4],[5,8,6,5]]
PD code (use to draw this multiloop with SnapPy): [[4,10,1,5],[5,11,6,16],[9,3,10,4],[1,12,2,11],[6,17,7,20],[15,19,16,20],[8,13,9,14],[2,12,3,13],[17,8,18,7],[18,14,19,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-11,-2)(2,11,-3,-12)(14,19,-15,-20)(20,7,-17,-8)(9,12,-10,-13)(10,3,-5,-4)(4,5,-1,-6)(15,6,-16,-7)(8,17,-9,-18)(18,13,-19,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,6)(-2,-12,9,17,7,-16)(-3,10,12)(-4,-6,15,19,13,-10)(-5,4)(-7,20,-15)(-8,-18,-14,-20)(-9,-13,18)(-11,2)(-17,8)(-19,14)(1,5,3,11)
Multiloop annotated with half-edges
12^4_1 annotated with half-edges