$12^{1}_{170}$ - Minimal pinning sets
Pinning sets for 12^1_170 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_170 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 556
of which optimal: 2
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.1133
on average over minimal pinning sets: 2.725
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)
• {2, 3, 5, 8}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 2, 5, 6}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {2, 3, 5, 6, 10}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {2, 3, 4, 5, 6}
5
[2, 2, 3, 3, 4]
2.80
c (minimal)
• {1, 2, 5, 8, 12}
5
[2, 2, 3, 3, 4]
2.80
d (minimal)
• {1, 2, 5, 8, 11}
5
[2, 2, 3, 3, 4]
2.80
e (minimal)
• {2, 3, 5, 6, 7}
5
[2, 2, 3, 3, 4]
2.80
f (minimal)
• {2, 3, 5, 6, 9}
5
[2, 2, 3, 3, 4]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.5
5
0
6
16
2.76
6
0
0
78
2.94
7
0
0
142
3.07
8
0
0
154
3.16
9
0
0
104
3.23
10
0
0
43
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
2
6
548
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4]
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],[0,8,6,5],[1,4,2,1],[2,4,9,3],[3,9,9,8],[3,7,9,4],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[10,17,11,18],[19,16,20,17],[12,3,13,4],[1,8,2,9],[18,9,19,10],[2,15,3,16],[13,6,14,7],[4,7,5,8],[5,14,6,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(11,4,-12,-5)(2,5,-3,-6)(19,6,-20,-7)(16,7,-17,-8)(3,12,-4,-13)(20,13,-1,-14)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,18,14)(-2,-6,19,-10)(-3,-13,20,6)(-4,11,1,13)(-5,2,-11)(-7,16,-9,-19)(-8,-16)(-12,3,5)(-14,17,7,-20)(-15,8,-17)(-18,9,15)(4,12)
Loop annotated with half-edges
12^1_170 annotated with half-edges