$12^{3}_{57}$ - Minimal pinning sets
Pinning sets for 12^3_57 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_57 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 332
of which optimal: 2
of which minimal: 10
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.11383
on average over minimal pinning sets: 2.77238
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, 2, 3, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 3, 5, 7, 12}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 2, 3, 8, 9, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
b (minimal)
• {1, 2, 3, 7, 8, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
c (minimal)
• {1, 3, 5, 8, 11, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
d (minimal)
• {1, 3, 5, 8, 9, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
e (minimal)
• {1, 3, 4, 8, 11, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
f (minimal)
• {1, 3, 4, 8, 9, 12}
6
[2, 2, 3, 3, 4, 4]
3.00
g (minimal)
• {1, 3, 4, 7, 8, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
h (minimal)
• {1, 2, 3, 5, 7, 10, 11}
7
[2, 2, 3, 3, 3, 3, 4]
2.86
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.6
6
0
7
14
2.81
7
0
1
66
2.99
8
0
0
103
3.11
9
0
0
87
3.2
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
2
8
322
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,7],[0,7,7,3],[0,2,8,4],[0,3,8,5],[1,4,9,6],[1,5,9,7],[1,6,2,2],[3,9,9,4],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[8,14,1,9],[9,5,10,6],[7,20,8,15],[13,19,14,20],[1,19,2,18],[4,17,5,18],[10,17,11,16],[6,16,7,15],[12,2,13,3],[3,11,4,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,2,-14,-3)(10,7,-11,-8)(8,9,-1,-10)(5,12,-6,-13)(17,14,-18,-9)(1,18,-2,-19)(19,6,-20,-7)(11,20,-12,-15)(4,15,-5,-16)(16,3,-17,-4)
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,-19,-7,10)(-2,13,-6,19)(-3,16,-5,-13)(-4,-16)(-8,-10)(-9,8,-11,-15,4,-17)(-12,5,15)(-14,17,3)(-18,1,9)(-20,11,7)(2,18,14)(6,12,20)
Multiloop annotated with half-edges
12^3_57 annotated with half-edges