$12^{1}_{338}$ - Minimal pinning sets
Pinning sets for 12^1_338 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_338 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 444
of which optimal: 2
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04947
on average over minimal pinning sets: 2.49167
on average over optimal pinning sets: 2.375
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 7, 9}
4
[2, 2, 2, 4]
2.50
B (optimal)
• {1, 3, 4, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 4, 6, 9}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {1, 4, 6, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
c (minimal)
• {1, 4, 6, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
d (minimal)
• {1, 4, 6, 8, 9}
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
2
0
0
2.38
5
0
4
15
2.64
6
0
0
63
2.85
7
0
0
111
3.0
8
0
0
120
3.1
9
0
0
83
3.18
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
4
438
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],[0,4,4,2],[0,1,5,6],[0,6,7,0],[1,7,5,1],[2,4,8,9],[2,9,7,3],[3,6,8,4],[5,7,9,9],[5,8,8,6]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,14,9,13],[19,12,20,13],[6,1,7,2],[14,10,15,9],[15,18,16,19],[11,2,12,3],[5,10,6,11],[17,4,18,5],[16,4,17,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,2,-16,-3)(4,9,-5,-10)(20,5,-1,-6)(14,7,-15,-8)(8,3,-9,-4)(10,13,-11,-14)(1,16,-2,-17)(6,17,-7,-18)(18,11,-19,-12)(12,19,-13,-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,-17,6)(-2,15,7,17)(-3,8,-15)(-4,-10,-14,-8)(-5,20,-13,10)(-6,-18,-12,-20)(-7,14,-11,18)(-9,4)(-16,1,5,9,3)(-19,12)(2,16)(11,13,19)
Loop annotated with half-edges
12^1_338 annotated with half-edges