$12^{1}_{168}$ - Minimal pinning sets
Pinning sets for 12^1_168 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_168 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 252
of which optimal: 6
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97224
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 8, 11}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 3, 4, 11}
5
[2, 2, 2, 2, 4]
2.40
D (optimal)
• {1, 2, 3, 5, 11}
5
[2, 2, 2, 2, 3]
2.20
E (optimal)
• {1, 2, 3, 10, 11}
5
[2, 2, 2, 2, 4]
2.40
F (optimal)
• {1, 2, 3, 9, 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
6
0
0
2.33
6
0
0
27
2.65
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
6
0
246
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,7],[0,7,8,8],[0,9,7,6],[1,6,6,1],[2,5,5,4],[2,4,9,3],[3,9,9,3],[4,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,15,1,16],[16,9,17,10],[10,19,11,20],[3,14,4,15],[1,6,2,7],[8,17,9,18],[18,7,19,8],[11,2,12,3],[13,4,14,5],[5,12,6,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(17,2,-18,-3)(14,3,-15,-4)(12,5,-13,-6)(1,8,-2,-9)(19,10,-20,-11)(16,11,-17,-12)(4,13,-5,-14)(6,15,-7,-16)(7,18,-8,-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,-9)(-2,17,11,-20,9)(-3,14,-5,12,-17)(-4,-14)(-6,-16,-12)(-7,-19,-11,16)(-8,1,-10,19)(-13,4,-15,6)(-18,7,15,3)(2,8,18)(5,13)(10,20)
Loop annotated with half-edges
12^1_168 annotated with half-edges