$12^{2}_{176}$ - Minimal pinning sets
Pinning sets for 12^2_176 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_176 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 382
of which optimal: 1
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03785
on average over minimal pinning sets: 2.55
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 7, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {4, 5, 6, 7, 11}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {4, 5, 7, 11, 12}
5
[2, 2, 2, 3, 4]
2.60
c (minimal)
• {4, 5, 7, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
d (minimal)
• {3, 4, 5, 7, 11}
5
[2, 2, 2, 3, 4]
2.60
e (minimal)
• {4, 5, 7, 8, 11}
5
[2, 2, 2, 3, 5]
2.80
f (minimal)
• {2, 4, 5, 7, 11}
5
[2, 2, 2, 3, 4]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
6
8
2.59
6
0
0
49
2.81
7
0
0
91
2.97
8
0
0
105
3.08
9
0
0
77
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
6
375
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],[0,5,6,7],[0,7,8,8],[0,8,9,6],[0,6,5,5],[1,4,4,6],[1,5,4,3],[1,9,9,2],[2,9,3,2],[3,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[16,7,1,8],[8,3,9,4],[15,20,16,17],[6,13,7,14],[1,10,2,11],[11,2,12,3],[9,12,10,13],[4,18,5,17],[19,14,20,15],[5,18,6,19]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,16,-10,-1)(6,1,-7,-2)(14,5,-15,-6)(15,8,-16,-9)(7,10,-8,-11)(4,11,-5,-12)(19,12,-20,-13)(3,18,-4,-19)(13,20,-14,-17)(17,2,-18,-3)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,-15,-9)(-2,17,-14,-6)(-3,-19,-13,-17)(-4,-12,19)(-5,14,20,12)(-7,-11,4,18,2)(-8,15,5,11)(-10,7,1)(-16,9)(-18,3)(-20,13)(8,10,16)
Multiloop annotated with half-edges
12^2_176 annotated with half-edges