$12^{1}_{162}$ - Minimal pinning sets
Pinning sets for 12^1_162 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_162 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 6, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 4, 5, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,7,7,4],[0,3,6,5],[1,4,2,1],[2,4,8,8],[3,9,9,3],[6,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,18,9,17],[19,16,20,17],[6,11,7,12],[1,11,2,10],[18,10,19,9],[2,15,3,16],[12,5,13,6],[14,3,15,4],[4,13,5,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(9,2,-10,-3)(17,4,-18,-5)(6,15,-7,-16)(7,18,-8,-19)(3,8,-4,-9)(1,10,-2,-11)(19,12,-20,-13)(16,13,-17,-14)(14,5,-15,-6)
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,-11)(-2,9,-4,17,13,-20,11)(-3,-9)(-5,14,-17)(-6,-16,-14)(-7,-19,-13,16)(-8,3,-10,1,-12,19)(-15,6)(-18,7,15,5)(2,10)(4,8,18)(12,20)
Loop annotated with half-edges
12^1_162 annotated with half-edges