$12^{1}_{262}$ - Minimal pinning sets
Pinning sets for 12^1_262 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_262 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 492
of which optimal: 2
of which minimal: 17
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.23247
on average over minimal pinning sets: 3.0
on average over optimal pinning sets: 3.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 6, 9, 11}
5
[3, 3, 3, 3, 3]
3.00
B (optimal)
• {1, 3, 5, 7, 10}
5
[3, 3, 3, 3, 3]
3.00
a (minimal)
• {2, 4, 6, 7, 10, 11}
6
[3, 3, 3, 3, 3, 3]
3.00
b (minimal)
• {2, 4, 5, 7, 10, 11}
6
[3, 3, 3, 3, 3, 3]
3.00
c (minimal)
• {2, 4, 5, 7, 9, 11}
6
[3, 3, 3, 3, 3, 3]
3.00
d (minimal)
• {2, 3, 5, 6, 9, 11}
6
[3, 3, 3, 3, 3, 3]
3.00
e (minimal)
• {2, 3, 5, 7, 10, 11}
6
[3, 3, 3, 3, 3, 3]
3.00
f (minimal)
• {2, 3, 5, 7, 9, 11}
6
[3, 3, 3, 3, 3, 3]
3.00
g (minimal)
• {1, 3, 5, 6, 9, 10}
6
[3, 3, 3, 3, 3, 3]
3.00
h (minimal)
• {1, 3, 4, 6, 9, 10}
6
[3, 3, 3, 3, 3, 3]
3.00
i (minimal)
• {1, 3, 4, 6, 7, 10}
6
[3, 3, 3, 3, 3, 3]
3.00
j (minimal)
• {1, 2, 4, 6, 9, 10}
6
[3, 3, 3, 3, 3, 3]
3.00
k (minimal)
• {1, 2, 4, 6, 7, 10}
6
[3, 3, 3, 3, 3, 3]
3.00
l (minimal)
• {1, 2, 4, 5, 7, 10}
6
[3, 3, 3, 3, 3, 3]
3.00
m (minimal)
• {1, 3, 5, 6, 9, 11}
6
[3, 3, 3, 3, 3, 3]
3.00
n (minimal)
• {1, 3, 5, 7, 9, 11}
6
[3, 3, 3, 3, 3, 3]
3.00
o (minimal)
• {1, 3, 4, 6, 9, 11}
6
[3, 3, 3, 3, 3, 3]
3.00
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
3.0
6
0
15
14
3.05
7
0
0
102
3.15
8
0
0
160
3.23
9
0
0
130
3.29
10
0
0
56
3.32
11
0
0
12
3.33
12
0
0
1
3.33
Total
2
15
475
Other information about this loop Properties
Region degree sequence: [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5]
Minimal region degree: 3
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,6,3],[0,2,7,4],[0,3,7,8],[1,8,9,6],[1,5,9,2],[3,9,8,4],[4,7,9,5],[5,8,7,6]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[5,12,6,13],[6,19,7,20],[14,7,15,8],[1,8,2,9],[17,4,18,5],[18,11,19,12],[15,3,16,2],[9,16,10,17],[10,3,11,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(14,1,-15,-2)(11,4,-12,-5)(18,5,-19,-6)(15,8,-16,-9)(2,9,-3,-10)(6,13,-7,-14)(3,16,-4,-17)(10,17,-11,-18)(19,12,-20,-13)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-7)(-2,-10,-18,-6,-14)(-3,-17,10)(-4,11,17)(-5,18,-11)(-8,15,1)(-9,2,-15)(-12,19,5)(-13,6,-19)(-16,3,9)(-20,7,13)(4,16,8,20,12)
Loop annotated with half-edges
12^1_262 annotated with half-edges