$12^{2}_{204}$ - Minimal pinning sets
Pinning sets for 12^2_204 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_204 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, 3, 7, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 7, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {1, 2, 7, 11, 12}
5
[2, 2, 2, 3, 5]
2.80
c (minimal)
• {1, 2, 6, 7, 11}
5
[2, 2, 2, 3, 5]
2.80
d (minimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 2, 3, 4]
2.60
e (minimal)
• {1, 2, 7, 8, 11}
5
[2, 2, 2, 3, 4]
2.60
f (minimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
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, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,4],[1,3,6,7],[2,7,8,8],[3,8,9,4],[4,9,9,5],[5,9,6,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[9,16,10,1],[15,8,16,9],[10,8,11,7],[1,7,2,6],[14,5,15,6],[11,17,12,20],[2,13,3,14],[4,17,5,18],[12,19,13,20],[3,19,4,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,3,-13,-4)(4,15,-5,-16)(5,8,-6,-9)(13,6,-14,-7)(16,11,-1,-12)(7,14,-8,-15)(18,1,-19,-2)(10,19,-11,-20)(20,9,-17,-10)(2,17,-3,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-3,12)(-2,-18)(-4,-16,-12)(-5,-9,20,-11,16)(-6,13,3,17,9)(-7,-15,4,-13)(-8,5,15)(-10,-20)(-14,7)(-17,2,-19,10)(1,11,19)(6,8,14)
Multiloop annotated with half-edges
12^2_204 annotated with half-edges