$12^{1}_{221}$ - Minimal pinning sets
Pinning sets for 12^1_221 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_221 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 168
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: 2.98097
on average over minimal pinning sets: 2.38889
on average over optimal pinning sets: 2.38889
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 7, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 3, 7, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 3, 5, 7, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
D (optimal)
• {1, 3, 5, 7, 8, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
E (optimal)
• {1, 3, 4, 7, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
F (optimal)
• {1, 3, 4, 7, 8, 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
6
6
0
0
2.39
7
0
0
27
2.71
8
0
0
50
2.93
9
0
0
49
3.08
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
6
0
162
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,6],[0,6,7,7],[0,8,9,9],[1,9,8,1],[2,7,3,2],[3,6,8,3],[4,7,5,9],[4,8,5,4]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[13,4,14,5],[14,19,15,20],[6,18,7,17],[1,11,2,10],[3,12,4,13],[18,15,19,16],[7,16,8,17],[11,8,12,9],[2,9,3,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(7,4,-8,-5)(18,5,-19,-6)(6,17,-7,-18)(19,8,-20,-9)(14,11,-15,-12)(3,12,-4,-13)(13,2,-14,-3)(10,15,-11,-16)(1,16,-2,-17)
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,-17,6,-19,-9)(-2,13,-4,7,17)(-3,-13)(-5,18,-7)(-6,-18)(-8,19,5)(-10,-16,1)(-11,14,2,16)(-12,3,-14)(-15,10,20,8,4,12)(-20,9)(11,15)
Loop annotated with half-edges
12^1_221 annotated with half-edges