$12^{1}_{107}$ - Minimal pinning sets
Pinning sets for 12^1_107 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_107 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 300
of which optimal: 2
of which minimal: 9
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05584
on average over minimal pinning sets: 2.59259
on average over optimal pinning sets: 2.5
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, 3, 4]
2.60
B (optimal)
• {1, 2, 4, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 3, 9, 11, 12}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 3, 8, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 2, 3, 6, 11, 12}
6
[2, 2, 2, 3, 3, 4]
2.67
d (minimal)
• {1, 2, 3, 6, 8, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
e (minimal)
• {1, 2, 4, 5, 11, 12}
6
[2, 2, 2, 3, 3, 4]
2.67
f (minimal)
• {1, 2, 4, 5, 7, 11}
6
[2, 2, 2, 3, 4, 4]
2.83
g (minimal)
• {1, 2, 3, 4, 11, 12}
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
2
0
0
2.5
6
0
7
14
2.7
7
0
0
64
2.91
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
2
7
291
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,3],[0,2,7,7],[0,5,1,1],[1,4,8,2],[2,8,9,7],[3,6,9,3],[5,9,9,6],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[2,9,3,10],[12,19,13,20],[4,13,5,14],[1,11,2,10],[11,8,12,9],[18,15,19,16],[5,15,6,14],[7,16,8,17],[17,6,18,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,4,-16,-5)(9,6,-10,-7)(20,7,-1,-8)(8,19,-9,-20)(2,11,-3,-12)(12,3,-13,-4)(16,13,-17,-14)(5,14,-6,-15)(10,17,-11,-18)(1,18,-2,-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,-19,8)(-2,-12,-4,15,-6,9,19)(-3,12)(-5,-15)(-7,20,-9)(-8,-20)(-10,-18,1,7)(-11,2,18)(-13,16,4)(-14,5,-16)(-17,10,6,14)(3,11,17,13)
Loop annotated with half-edges
12^1_107 annotated with half-edges