$12^{1}_{166}$ - Minimal pinning sets
Pinning sets for 12^1_166 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_166 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 224
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: 2.9785
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 4, 8, 11}
5
[2, 2, 2, 2, 3]
2.20
C (optimal)
• {1, 3, 4, 7, 11}
5
[2, 2, 2, 2, 4]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.27
6
0
0
18
2.59
7
0
0
46
2.82
8
0
0
65
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
0
221
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,5,6,0],[0,7,8,4],[1,3,5,1],[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): [[9,20,10,1],[13,8,14,9],[19,10,20,11],[1,6,2,7],[7,12,8,13],[14,12,15,11],[3,18,4,19],[5,16,6,17],[2,16,3,15],[17,4,18,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,20,-14,-1)(17,2,-18,-3)(9,4,-10,-5)(15,6,-16,-7)(7,10,-8,-11)(3,8,-4,-9)(11,18,-12,-19)(1,12,-2,-13)(19,14,-20,-15)(5,16,-6,-17)
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,-13)(-2,17,-6,15,-20,13)(-3,-9,-5,-17)(-4,9)(-7,-11,-19,-15)(-8,3,-18,11)(-10,7,-16,5)(-12,1,-14,19)(2,12,18)(4,8,10)(6,16)(14,20)
Loop annotated with half-edges
12^1_166 annotated with half-edges