$12^{1}_{326}$ - Minimal pinning sets
Pinning sets for 12^1_326 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_326 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 528
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: 3.12437
on average over minimal pinning sets: 2.675
on average over optimal pinning sets: 2.625
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 8}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 3, 8, 10}
4
[2, 2, 3, 4]
2.75
a (minimal)
• {1, 3, 7, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 3, 5, 8, 9}
5
[2, 2, 3, 3, 4]
2.80
c (minimal)
• {1, 3, 4, 7, 9}
5
[2, 2, 3, 3, 3]
2.60
d (minimal)
• {1, 3, 6, 7, 9}
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
2
0
0
2.62
5
0
4
15
2.81
6
0
0
69
2.96
7
0
0
132
3.08
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
2
4
522
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],[0,5,6,2],[0,1,7,3],[0,2,8,4],[0,3,9,5],[1,4,6,6],[1,5,5,7],[2,6,9,8],[3,7,9,9],[4,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,5,1,6],[6,9,7,10],[10,19,11,20],[11,4,12,5],[1,12,2,13],[13,8,14,9],[7,14,8,15],[15,18,16,19],[16,3,17,4],[2,17,3,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(9,2,-10,-3)(18,3,-19,-4)(11,6,-12,-7)(4,7,-5,-8)(19,10,-20,-11)(5,12,-6,-13)(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)(-2,9,17,-14)(-3,18,-9)(-4,-8,-18)(-5,-13,-17,8)(-6,11,-20,-16,13)(-7,4,-19,-11)(-10,19,3)(-12,5,7)(-15,20,10,2)(1,15)(6,12)
Loop annotated with half-edges
12^1_326 annotated with half-edges