$12^{1}_{59}$ - Minimal pinning sets
Pinning sets for 12^1_59 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_59 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 256
of which optimal: 1
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04981
on average over minimal pinning sets: 2.59167
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 6, 8}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 3, 6, 8, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 3, 6, 7, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 3, 4, 6, 8, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
d (minimal)
• {1, 3, 4, 6, 7, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
e (minimal)
• {1, 3, 5, 6, 7, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
f (minimal)
• {1, 2, 3, 6, 8, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
g (minimal)
• {1, 3, 4, 6, 8, 9}
6
[2, 2, 2, 3, 4, 4]
2.83
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
2.4
6
0
7
7
2.64
7
0
0
49
2.87
8
0
0
79
3.04
9
0
0
69
3.15
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
7
248
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: [[1,2,2,3],[0,3,4,5],[0,6,3,0],[0,2,6,1],[1,6,7,5],[1,4,8,8],[2,9,4,3],[4,9,9,8],[5,7,9,5],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[6,17,7,18],[19,8,20,9],[1,19,2,18],[10,5,11,6],[11,16,12,17],[9,3,10,2],[4,13,5,14],[15,12,16,13],[3,15,4,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,2,-8,-3)(16,5,-17,-6)(12,9,-13,-10)(1,10,-2,-11)(11,20,-12,-1)(8,13,-9,-14)(3,14,-4,-15)(15,18,-16,-19)(4,17,-5,-18)(19,6,-20,-7)
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,-11)(-2,7,-20,11)(-3,-15,-19,-7)(-4,-18,15)(-5,16,18)(-6,19,-16)(-8,-14,3)(-9,12,20,6,-17,4,14)(-10,1,-12)(-13,8,2,10)(5,17)(9,13)
Loop annotated with half-edges
12^1_59 annotated with half-edges