$12^{1}_{216}$ - Minimal pinning sets
Pinning sets for 12^1_216 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_216 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 190
of which optimal: 1
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97295
on average over minimal pinning sets: 2.42222
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 5, 8, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {2, 3, 4, 5, 6, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
b (minimal)
• {2, 3, 5, 6, 7, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
c (minimal)
• {2, 3, 5, 6, 10, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
d (minimal)
• {2, 3, 5, 6, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
e (minimal)
• {1, 2, 3, 5, 6, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
5
7
2.5
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
5
184
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 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],[0,6,7,8],[0,8,8,5],[1,4,2,1],[2,9,9,3],[3,9,9,8],[3,7,4,4],[6,7,7,6]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[10,17,11,18],[19,16,20,17],[12,4,13,3],[1,8,2,9],[18,9,19,10],[15,4,16,5],[13,6,14,7],[7,2,8,3],[5,14,6,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,20,-6,-1)(1,10,-2,-11)(13,2,-14,-3)(11,4,-12,-5)(19,6,-20,-7)(16,7,-17,-8)(3,12,-4,-13)(17,14,-18,-15)(8,15,-9,-16)(9,18,-10,-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,-11,-5)(-2,13,-4,11)(-3,-13)(-6,19,-10,1)(-7,16,-9,-19)(-8,-16)(-12,3,-14,17,7,-20,5)(-15,8,-17)(-18,9,15)(2,10,18,14)(4,12)(6,20)
Loop annotated with half-edges
12^1_216 annotated with half-edges