$12^{1}_{280}$ - Minimal pinning sets
Pinning sets for 12^1_280 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_280 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 264
of which optimal: 1
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10067
on average over minimal pinning sets: 2.68435
on average over optimal pinning sets: 2.6
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 10}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 2, 4, 6, 10, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 2, 4, 5, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
c (minimal)
• {1, 2, 4, 5, 10, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
d (minimal)
• {1, 2, 3, 4, 6, 10}
6
[2, 2, 3, 3, 3, 3]
2.67
e (minimal)
• {1, 2, 3, 5, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
f (minimal)
• {1, 2, 4, 6, 8, 9, 11}
7
[2, 2, 3, 3, 3, 3, 4]
2.86
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
2.6
6
0
5
7
2.75
7
0
1
44
2.93
8
0
0
80
3.07
9
0
0
76
3.18
10
0
0
39
3.26
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
6
257
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,6,3],[0,2,7,7],[0,8,9,5],[1,4,9,6],[1,5,9,2],[3,8,8,3],[4,7,7,9],[4,8,6,5]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[11,4,12,5],[12,19,13,20],[6,13,7,14],[1,16,2,17],[17,10,18,11],[18,3,19,4],[7,15,8,14],[8,15,9,16],[2,9,3,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,20,-16,-1)(8,1,-9,-2)(19,6,-20,-7)(14,7,-15,-8)(3,10,-4,-11)(11,4,-12,-5)(5,12,-6,-13)(18,13,-19,-14)(9,16,-10,-17)(2,17,-3,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,-15)(-2,-18,-14,-8)(-3,-11,-5,-13,18)(-4,11)(-6,19,13)(-7,14,-19)(-9,-17,2)(-10,3,17)(-12,5)(-16,9,1)(-20,15,7)(4,10,16,20,6,12)
Loop annotated with half-edges
12^1_280 annotated with half-edges