$12^{3}_{70}$ - Minimal pinning sets
Pinning sets for 12^3_70 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_70 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 700
of which optimal: 2
of which minimal: 17
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.19712
on average over minimal pinning sets: 2.98725
on average over optimal pinning sets: 2.875
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 7, 10, 11}
4
[2, 3, 3, 3]
2.75
B (optimal)
• {1, 4, 6, 9}
4
[2, 3, 3, 4]
3.00
a (minimal)
• {1, 7, 9, 11, 12}
5
[2, 3, 3, 3, 4]
3.00
b (minimal)
• {1, 6, 7, 9, 12}
5
[2, 3, 3, 3, 4]
3.00
c (minimal)
• {1, 6, 8, 9, 12}
5
[2, 3, 3, 4, 4]
3.20
d (minimal)
• {1, 6, 8, 9, 11}
5
[2, 3, 3, 3, 4]
3.00
e (minimal)
• {1, 7, 8, 9, 11}
5
[2, 3, 3, 3, 4]
3.00
f (minimal)
• {1, 4, 7, 9, 11}
5
[2, 3, 3, 3, 4]
3.00
g (minimal)
• {1, 2, 6, 9, 12}
5
[2, 3, 3, 3, 4]
3.00
h (minimal)
• {1, 3, 6, 8, 9}
5
[2, 3, 3, 3, 4]
3.00
i (minimal)
• {1, 2, 3, 6, 9}
5
[2, 3, 3, 3, 3]
2.80
j (minimal)
• {1, 5, 7, 9, 11}
5
[2, 3, 3, 3, 5]
3.20
k (minimal)
• {1, 2, 6, 9, 10, 11}
6
[2, 3, 3, 3, 3, 3]
2.83
l (minimal)
• {1, 3, 6, 7, 9, 10}
6
[2, 3, 3, 3, 3, 3]
2.83
m (minimal)
• {1, 2, 3, 7, 9, 11}
6
[2, 3, 3, 3, 3, 3]
2.83
n (minimal)
• {1, 2, 5, 6, 9, 11}
6
[2, 3, 3, 3, 3, 5]
3.17
o (minimal)
• {1, 3, 5, 6, 7, 9}
6
[2, 3, 3, 3, 3, 5]
3.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.88
5
0
10
16
3.02
6
0
5
99
3.1
7
0
0
186
3.17
8
0
0
194
3.23
9
0
0
126
3.27
10
0
0
50
3.3
11
0
0
11
3.32
12
0
0
1
3.33
Total
2
15
683
Other information about this multiloop Properties
Region degree sequence: [2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,3],[0,2,6,4],[0,3,7,1],[1,7,6,2],[3,5,8,9],[4,9,8,5],[6,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,17,10,20],[7,19,8,20],[15,18,16,19],[1,18,2,17],[10,6,11,7],[11,14,12,15],[2,5,3,6],[3,13,4,14],[12,4,13,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,2,-10,-3)(8,3,-1,-4)(15,4,-16,-5)(1,10,-2,-11)(16,11,-9,-12)(6,13,-7,-14)(12,17,-13,-18)(5,18,-6,-19)(19,14,-20,-15)(20,7,-17,-8)
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,-11,16,4)(-2,9,11)(-3,8,-17,12,-9)(-4,15,-20,-8)(-5,-19,-15)(-6,-14,19)(-7,20,14)(-10,1,3)(-12,-18,5,-16)(-13,6,18)(2,10)(7,13,17)
Multiloop annotated with half-edges
12^3_70 annotated with half-edges