$12^{1}_{227}$ - Minimal pinning sets
Pinning sets for 12^1_227 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_227 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 272
of which optimal: 3
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0386
on average over minimal pinning sets: 2.46667
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 5, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 3, 5, 8}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 3, 5, 7, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
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
2.4
6
0
1
19
2.68
7
0
0
55
2.89
8
0
0
81
3.04
9
0
0
69
3.15
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
1
268
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,2,2],[0,1,1,3],[0,2,6,7],[0,7,8,5],[1,4,8,9],[3,9,7,7],[3,6,6,4],[4,9,9,5],[5,8,8,6]]
PD code (use to draw this loop with SnapPy): [[20,17,1,18],[18,11,19,12],[12,19,13,20],[13,16,14,17],[1,6,2,7],[7,10,8,11],[4,15,5,16],[14,5,15,6],[2,9,3,10],[8,3,9,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(12,3,-13,-4)(19,4,-20,-5)(20,7,-1,-8)(5,8,-6,-9)(16,9,-17,-10)(2,13,-3,-14)(11,14,-12,-15)(18,15,-19,-16)(10,17,-11,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,8)(-2,-14,11,17,9,-6)(-3,12,14)(-4,19,15,-12)(-5,-9,16,-19)(-7,20,4,-13,2)(-8,5,-20)(-10,-18,-16)(-11,-15,18)(-17,10)(1,7)(3,13)
Loop annotated with half-edges
12^1_227 annotated with half-edges