$12^{1}_{79}$ - Minimal pinning sets
Pinning sets for 12^1_79 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_79 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 456
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: 3.05346
on average over minimal pinning sets: 2.525
on average over optimal pinning sets: 2.375
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 7, 11}
4
[2, 2, 2, 4]
2.50
B (optimal)
• {1, 2, 3, 7}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 5, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {1, 3, 5, 7, 9}
5
[2, 2, 2, 3, 4]
2.60
c (minimal)
• {1, 3, 4, 7, 8}
5
[2, 2, 2, 3, 4]
2.60
d (minimal)
• {1, 3, 4, 7, 9}
5
[2, 2, 2, 4, 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
2.38
5
0
4
15
2.65
6
0
0
65
2.86
7
0
0
116
3.0
8
0
0
124
3.11
9
0
0
84
3.19
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
4
450
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],[0,3,2,0],[0,1,4,5],[0,5,6,1],[2,7,8,8],[2,8,9,3],[3,9,7,7],[4,6,6,9],[4,9,5,4],[5,8,7,6]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[19,12,20,13],[14,12,15,11],[1,18,2,19],[15,4,16,5],[17,10,18,11],[2,7,3,8],[8,3,9,4],[16,6,17,5],[6,9,7,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(13,2,-14,-3)(3,14,-4,-15)(20,5,-1,-6)(4,7,-5,-8)(15,8,-16,-9)(9,12,-10,-13)(17,10,-18,-11)(19,16,-20,-17)(11,18,-12,-19)
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)(-2,13,-10,17,-20,-6)(-3,-15,-9,-13)(-4,-8,15)(-5,20,16,8)(-7,4,14,2)(-11,-19,-17)(-12,9,-16,19)(-14,3)(-18,11)(1,5,7)(10,12,18)
Loop annotated with half-edges
12^1_79 annotated with half-edges