$12^{1}_{282}$ - Minimal pinning sets
Pinning sets for 12^1_282 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_282 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 344
of which optimal: 2
of which minimal: 10
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.11364
on average over minimal pinning sets: 2.75333
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, 3, 4, 7, 10}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 4, 5, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 3, 4, 5, 9, 10}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 4, 5, 7, 10, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
c (minimal)
• {1, 4, 6, 7, 10, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
d (minimal)
• {1, 4, 6, 7, 9, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
e (minimal)
• {1, 3, 4, 7, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
f (minimal)
• {1, 2, 4, 5, 9, 10}
6
[2, 2, 3, 3, 3, 4]
2.83
g (minimal)
• {1, 2, 4, 5, 7, 10}
6
[2, 2, 3, 3, 3, 4]
2.83
h (minimal)
• {1, 2, 4, 6, 7, 10}
6
[2, 2, 3, 3, 4, 4]
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
2.6
6
0
8
14
2.8
7
0
0
71
2.98
8
0
0
108
3.12
9
0
0
89
3.21
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
2
8
334
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,7],[0,7,7,3],[0,2,8,4],[0,3,8,5],[1,4,8,9],[1,9,9,7],[1,6,2,2],[3,9,5,4],[5,8,6,6]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[17,12,18,13],[6,19,7,20],[14,7,15,8],[1,8,2,9],[9,16,10,17],[4,11,5,12],[18,5,19,6],[15,3,16,2],[10,3,11,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(15,2,-16,-3)(11,4,-12,-5)(18,5,-19,-6)(1,8,-2,-9)(14,9,-15,-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,-9,14,-7)(-2,15,9)(-3,-17,10,-15)(-4,11,17)(-5,18,-11)(-6,-14,-10,-18)(-8,1)(-12,19,5)(-13,6,-19)(-16,3)(-20,7,13)(2,8,20,12,4,16)
Loop annotated with half-edges
12^1_282 annotated with half-edges