$12^{1}_{333}$ - Minimal pinning sets
Pinning sets for 12^1_333 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_333 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 448
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.11448
on average over minimal pinning sets: 2.66
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 10, 12}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 3, 7, 9, 12}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {1, 2, 3, 7, 12}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {1, 3, 6, 10, 11}
5
[2, 2, 3, 3, 3]
2.60
d (minimal)
• {1, 3, 5, 10, 11}
5
[2, 2, 3, 3, 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
1
0
0
2.5
5
0
4
8
2.73
6
0
0
50
2.91
7
0
0
106
3.05
8
0
0
130
3.14
9
0
0
96
3.22
10
0
0
42
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
4
443
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,6],[0,7,8,3],[0,2,9,4],[0,3,9,5],[1,4,9,6],[1,5,7,1],[2,6,8,8],[2,7,7,9],[3,8,5,4]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[11,6,12,7],[19,16,20,17],[8,16,9,15],[1,15,2,14],[10,13,11,14],[5,12,6,13],[17,5,18,4],[18,3,19,4],[9,3,10,2]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,7,-1,-8)(16,1,-17,-2)(2,15,-3,-16)(9,4,-10,-5)(3,10,-4,-11)(8,11,-9,-12)(12,19,-13,-20)(13,6,-14,-7)(17,14,-18,-15)(5,18,-6,-19)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-3,-11,8)(-2,-16)(-4,9,11)(-5,-19,12,-9)(-6,13,19)(-7,20,-13)(-8,-12,-20)(-10,3,15,-18,5)(-14,17,1,7)(-15,2,-17)(4,10)(6,18,14)
Loop annotated with half-edges
12^1_333 annotated with half-edges