$12^{1}_{200}$ - Minimal pinning sets
Pinning sets for 12^1_200 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_200 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 386
of which optimal: 3
of which minimal: 12
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.12118
on average over minimal pinning sets: 2.79167
on average over optimal pinning sets: 2.66667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 10, 12}
5
[2, 2, 3, 3, 4]
2.80
B (optimal)
• {1, 3, 5, 8, 12}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 3, 5, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 3, 4, 8, 9, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 3, 4, 7, 10, 12}
6
[2, 2, 3, 3, 4, 4]
3.00
c (minimal)
• {1, 3, 4, 7, 8, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
d (minimal)
• {1, 3, 4, 6, 9, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
e (minimal)
• {1, 3, 4, 8, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
f (minimal)
• {1, 3, 4, 7, 9, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
g (minimal)
• {1, 3, 5, 7, 11, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
h (minimal)
• {1, 3, 4, 7, 11, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
i (minimal)
• {1, 3, 4, 6, 9, 10, 12}
7
[2, 2, 3, 3, 3, 4, 4]
3.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.67
6
0
8
20
2.85
7
0
1
84
3.01
8
0
0
122
3.13
9
0
0
95
3.22
10
0
0
42
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
3
9
374
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 6]
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],[2,8,9,6],[3,5,9,4],[4,9,8,8],[5,7,7,9],[5,8,7,6]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,19,11,20],[11,8,12,9],[1,12,2,13],[13,18,14,19],[16,7,17,8],[2,17,3,18],[14,6,15,5],[15,4,16,5],[6,3,7,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(1,10,-2,-11)(12,3,-13,-4)(7,4,-8,-5)(16,5,-17,-6)(17,8,-18,-9)(18,13,-19,-14)(9,14,-10,-15)(6,15,-7,-16)(2,19,-3,-20)
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)(-2,-20,11)(-3,12,20)(-4,7,15,-10,1,-12)(-5,16,-7)(-6,-16)(-8,17,5)(-9,-15,6,-17)(-13,18,8,4)(-14,9,-18)(-19,2,10,14)(3,19,13)
Loop annotated with half-edges
12^1_200 annotated with half-edges