$12^{1}_{128}$ - Minimal pinning sets
Pinning sets for 12^1_128 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_128 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, 2, 5, 6, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 3, 5, 6, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 5, 7, 11}
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, 3, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,6,3],[0,2,6,7],[0,5,1,1],[1,4,8,9],[2,9,3,2],[3,9,8,8],[5,7,7,9],[5,8,7,6]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[2,11,3,12],[6,19,7,20],[4,7,5,8],[1,13,2,12],[13,10,14,11],[18,5,19,6],[8,15,9,16],[16,9,17,10],[14,17,15,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,3,-17,-4)(2,5,-3,-6)(14,7,-15,-8)(11,8,-12,-9)(20,9,-1,-10)(10,19,-11,-20)(6,13,-7,-14)(12,15,-13,-16)(4,17,-5,-18)(1,18,-2,-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,-19,10)(-2,-6,-14,-8,11,19)(-3,16,-13,6)(-4,-18,1,9,-12,-16)(-5,2,18)(-7,14)(-9,20,-11)(-10,-20)(-15,12,8)(-17,4)(3,5,17)(7,13,15)
Loop annotated with half-edges
12^1_128 annotated with half-edges