$12^{1}_{342}$ - Minimal pinning sets
Pinning sets for 12^1_342 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_342 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 520
of which optimal: 1
of which minimal: 9
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.12284
on average over minimal pinning sets: 2.74444
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, 5, 10}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 2, 3, 8, 10}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 2, 3, 8, 12}
5
[2, 2, 3, 3, 4]
2.80
c (minimal)
• {1, 3, 8, 9, 12}
5
[2, 2, 3, 4, 4]
3.00
d (minimal)
• {1, 3, 8, 10, 12}
5
[2, 2, 3, 3, 4]
2.80
e (minimal)
• {1, 2, 3, 4, 8}
5
[2, 2, 3, 3, 3]
2.60
f (minimal)
• {1, 3, 4, 8, 9}
5
[2, 2, 3, 3, 4]
2.80
g (minimal)
• {1, 3, 4, 8, 10}
5
[2, 2, 3, 3, 3]
2.60
h (minimal)
• {1, 3, 6, 8, 10}
5
[2, 2, 3, 3, 5]
3.00
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
8
8
2.76
6
0
0
66
2.94
7
0
0
131
3.07
8
0
0
149
3.16
9
0
0
103
3.23
10
0
0
43
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
8
511
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,6],[0,6,6,7],[0,7,8,1],[1,8,9,2],[2,7,3,3],[3,6,9,4],[4,9,9,5],[5,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,13,1,14],[14,9,15,10],[10,19,11,20],[12,5,13,6],[1,8,2,9],[15,18,16,19],[11,7,12,6],[7,4,8,5],[2,17,3,18],[16,3,17,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(5,2,-6,-3)(10,3,-11,-4)(4,9,-5,-10)(19,6,-20,-7)(12,7,-13,-8)(18,11,-19,-12)(20,15,-1,-16)(13,16,-14,-17)(8,17,-9,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,16)(-2,5,9,17,-14)(-3,10,-5)(-4,-10)(-6,19,11,3)(-7,12,-19)(-8,-18,-12)(-9,4,-11,18)(-13,-17,8)(-15,20,6,2)(-16,13,7,-20)(1,15)
Loop annotated with half-edges
12^1_342 annotated with half-edges