$12^{1}_{137}$ - Minimal pinning sets
Pinning sets for 12^1_137 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_137 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 244
of which optimal: 1
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0456
on average over minimal pinning sets: 2.58435
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 8}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 5, 8, 12}
6
[2, 2, 2, 3, 3, 4]
2.67
b (minimal)
• {1, 2, 4, 5, 8, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 2, 4, 5, 7, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
d (minimal)
• {1, 2, 3, 5, 7, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
e (minimal)
• {1, 2, 3, 5, 7, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
f (minimal)
• {1, 2, 4, 5, 7, 10, 12}
7
[2, 2, 2, 3, 3, 4, 4]
2.86
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.4
6
0
5
7
2.63
7
0
1
43
2.85
8
0
0
75
3.02
9
0
0
68
3.15
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
6
237
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,3,0],[0,2,7,1],[1,7,8,5],[1,4,8,8],[2,9,9,7],[3,6,9,4],[4,9,5,5],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[10,17,11,18],[19,12,20,13],[1,19,2,18],[6,9,7,10],[7,16,8,17],[13,4,14,5],[2,5,3,6],[15,8,16,9],[3,14,4,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(19,2,-20,-3)(12,5,-13,-6)(3,6,-4,-7)(16,9,-17,-10)(20,11,-1,-12)(4,13,-5,-14)(7,14,-8,-15)(15,18,-16,-19)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-17,8,14,-5,12)(-2,19,-16,-10)(-3,-7,-15,-19)(-4,-14,7)(-6,3,-20,-12)(-8,-18,15)(-9,16,18)(-11,20,2)(-13,4,6)(1,11)(5,13)(9,17)
Loop annotated with half-edges
12^1_137 annotated with half-edges