$12^{1}_{336}$ - Minimal pinning sets
Pinning sets for 12^1_336 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_336 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 476
of which optimal: 3
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04838
on average over minimal pinning sets: 2.56667
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 7}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 5, 6, 7}
4
[2, 2, 2, 4]
2.50
C (optimal)
• {1, 5, 7, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 5, 7, 8, 10}
5
[2, 2, 2, 4, 4]
2.80
b (minimal)
• {1, 3, 5, 7, 10}
5
[2, 2, 2, 4, 4]
2.80
c (minimal)
• {1, 5, 7, 10, 11}
5
[2, 2, 2, 4, 4]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
3
0
0
2.33
5
0
3
21
2.67
6
0
0
73
2.87
7
0
0
121
3.01
8
0
0
125
3.11
9
0
0
84
3.19
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
3
470
Other information about this loop 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: [[1,2,3,3],[0,4,5,6],[0,6,6,7],[0,7,4,0],[1,3,7,8],[1,8,9,6],[1,5,2,2],[2,9,4,3],[4,9,9,5],[5,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,17,9,18],[14,19,15,20],[6,1,7,2],[16,5,17,6],[9,12,10,13],[18,13,19,14],[15,3,16,2],[11,4,12,5],[10,4,11,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,7,-1,-8)(8,1,-9,-2)(13,2,-14,-3)(17,4,-18,-5)(6,9,-7,-10)(15,10,-16,-11)(5,14,-6,-15)(11,16,-12,-17)(3,18,-4,-19)(19,12,-20,-13)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,13,-20,-8)(-3,-19,-13)(-4,17,-12,19)(-5,-15,-11,-17)(-6,-10,15)(-7,20,12,16,10)(-9,6,14,2)(-14,5,-18,3)(-16,11)(1,7,9)(4,18)
Loop annotated with half-edges
12^1_336 annotated with half-edges