$12^{1}_{110}$ - Minimal pinning sets
Pinning sets for 12^1_110 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_110 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 472
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: 3.10894
on average over minimal pinning sets: 2.67143
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 12}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 3, 4, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 2, 3, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {1, 3, 4, 11, 12}
5
[2, 2, 3, 3, 3]
2.60
d (minimal)
• {1, 3, 4, 6, 12}
5
[2, 2, 3, 3, 4]
2.80
e (minimal)
• {1, 3, 4, 8, 12}
5
[2, 2, 3, 3, 3]
2.60
f (minimal)
• {1, 3, 4, 10, 12}
5
[2, 2, 3, 3, 5]
3.00
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
2.5
5
0
6
8
2.73
6
0
0
57
2.91
7
0
0
115
3.05
8
0
0
135
3.14
9
0
0
97
3.22
10
0
0
42
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
6
465
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,6,7,3],[0,2,8,9],[0,6,5,1],[1,4,6,1],[2,5,4,9],[2,9,8,8],[3,7,7,9],[3,8,7,6]]
PD code (use to draw this loop with SnapPy): [[15,20,16,1],[14,3,15,4],[6,19,7,20],[16,7,17,8],[1,5,2,4],[2,13,3,14],[5,12,6,13],[9,18,10,19],[17,10,18,11],[8,11,9,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(2,19,-3,-20)(14,5,-15,-6)(15,8,-16,-9)(6,9,-7,-10)(3,10,-4,-11)(20,11,-1,-12)(7,16,-8,-17)(4,17,-5,-18)(13,18,-14,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12)(-2,-20,-12)(-3,-11,20)(-4,-18,13,1,11)(-5,14,18)(-6,-10,3,19,-14)(-7,-17,4,10)(-8,15,5,17)(-9,6,-15)(-13,-19,2)(-16,7,9)(8,16)
Loop annotated with half-edges
12^1_110 annotated with half-edges