$12^{1}_{116}$ - Minimal pinning sets
Pinning sets for 12^1_116 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_116 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 448
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04602
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 9, 10}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {2, 3, 7, 9}
4
[2, 2, 2, 3]
2.25
C (optimal)
• {2, 3, 9, 11}
4
[2, 2, 2, 4]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
3
0
0
2.33
5
0
0
21
2.65
6
0
0
64
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
3
0
445
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,3],[0,2,7,8],[0,5,1,1],[1,4,8,2],[2,8,9,9],[3,9,9,8],[3,7,6,5],[6,7,7,6]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[13,2,14,3],[19,10,20,11],[4,10,5,9],[1,12,2,13],[14,12,15,11],[18,7,19,8],[5,17,6,16],[8,15,9,16],[6,17,7,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,4,-16,-5)(6,1,-7,-2)(16,7,-17,-8)(8,11,-9,-12)(18,9,-19,-10)(5,12,-6,-13)(13,2,-14,-3)(3,14,-4,-15)(20,17,-1,-18)(10,19,-11,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,12,-9,18)(-2,13,-6)(-3,-15,-5,-13)(-4,15)(-7,16,4,14,2)(-8,-12,5,-16)(-10,-20,-18)(-11,8,-17,20)(-14,3)(-19,10)(1,17,7)(9,11,19)
Loop annotated with half-edges
12^1_116 annotated with half-edges