$12^{1}_{212}$ - Minimal pinning sets
Pinning sets for 12^1_212 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_212 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 304
of which optimal: 3
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05353
on average over minimal pinning sets: 2.58095
on average over optimal pinning sets: 2.46667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 6, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 3, 6, 10, 11}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 3, 4, 5, 10, 11}
6
[2, 2, 2, 3, 4, 4]
2.83
b (minimal)
• {1, 3, 4, 5, 9, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
c (minimal)
• {1, 3, 5, 6, 9, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
d (minimal)
• {1, 3, 6, 7, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
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
2.47
6
0
4
19
2.72
7
0
0
65
2.92
8
0
0
93
3.06
9
0
0
75
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
4
297
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,7,4],[0,3,8,9],[1,9,2,1],[2,7,7,3],[3,6,6,8],[4,7,9,9],[4,8,8,5]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,18,11,17],[19,16,20,17],[8,5,9,6],[1,5,2,4],[18,12,19,11],[15,6,16,7],[7,14,8,15],[2,14,3,13],[3,12,4,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,18,-2,-19)(5,2,-6,-3)(12,3,-13,-4)(17,6,-18,-7)(8,15,-9,-16)(20,9,-1,-10)(10,19,-11,-20)(4,11,-5,-12)(16,13,-17,-14)(14,7,-15,-8)
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)(-2,5,11,19)(-3,12,-5)(-4,-12)(-6,17,13,3)(-7,14,-17)(-8,-16,-14)(-9,20,-11,4,-13,16)(-10,-20)(-15,8)(-18,1,9,15,7)(2,18,6)
Loop annotated with half-edges
12^1_212 annotated with half-edges