$12^{1}_{201}$ - Minimal pinning sets
Pinning sets for 12^1_201 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_201 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 356
of which optimal: 1
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0457
on average over minimal pinning sets: 2.58611
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 3, 6, 9}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {1, 2, 3, 5, 9, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
c (minimal)
• {1, 2, 3, 5, 8, 9}
6
[2, 2, 2, 3, 4, 4]
2.83
d (minimal)
• {1, 2, 3, 7, 9, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
e (minimal)
• {1, 2, 3, 7, 8, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
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
2.25
5
0
1
8
2.58
6
0
4
34
2.79
7
0
0
83
2.95
8
0
0
103
3.08
9
0
0
77
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
5
350
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,4],[1,3,6,7],[2,8,8,6],[3,5,9,4],[4,9,9,8],[5,7,9,5],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[19,8,20,9],[10,8,11,7],[1,7,2,6],[18,5,19,6],[11,16,12,17],[2,17,3,18],[4,13,5,14],[15,12,16,13],[3,15,4,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,20,-14,-1)(1,16,-2,-17)(2,5,-3,-6)(14,3,-15,-4)(10,7,-11,-8)(19,8,-20,-9)(9,18,-10,-19)(6,11,-7,-12)(17,12,-18,-13)(4,15,-5,-16)
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,-13)(-2,-6,-12,17)(-3,14,20,8,-11,6)(-4,-16,1,-14)(-5,2,16)(-7,10,18,12)(-8,19,-10)(-9,-19)(-15,4)(-18,9,-20,13)(3,5,15)(7,11)
Loop annotated with half-edges
12^1_201 annotated with half-edges