$12^{1}_{50}$ - Minimal pinning sets
Pinning sets for 12^1_50 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_50 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 218
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: 2.99219
on average over minimal pinning sets: 2.46964
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)
• {2, 4, 6, 9, 11}
5
[2, 2, 2, 2, 4]
2.40
a (minimal)
• {2, 3, 4, 8, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {2, 4, 7, 8, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
c (minimal)
• {1, 2, 4, 8, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
d (minimal)
• {1, 2, 3, 4, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
e (minimal)
• {1, 2, 4, 5, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
f (minimal)
• {2, 3, 4, 9, 10, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
g (minimal)
• {2, 4, 5, 7, 9, 10, 11}
7
[2, 2, 2, 2, 4, 4, 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
2.4
6
0
6
7
2.55
7
0
1
43
2.8
8
0
0
67
2.99
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
7
210
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,7,7],[0,7,8,8],[0,9,9,1],[1,9,9,6],[1,5,8,2],[2,8,3,2],[3,7,6,3],[4,5,5,4]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[4,9,5,10],[14,19,15,20],[6,18,7,17],[1,11,2,10],[12,3,13,4],[13,8,14,9],[18,15,19,16],[7,16,8,17],[11,3,12,2]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(10,5,-11,-6)(1,6,-2,-7)(7,18,-8,-19)(15,12,-16,-13)(4,13,-5,-14)(14,3,-15,-4)(11,16,-12,-17)(2,17,-3,-18)(19,8,-20,-9)
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,-7,-19,-9)(-2,-18,7)(-3,14,-5,10,20,8,18)(-4,-14)(-6,1,-10)(-8,19)(-11,-17,2,6)(-12,15,3,17)(-13,4,-15)(-16,11,5,13)(-20,9)(12,16)
Loop annotated with half-edges
12^1_50 annotated with half-edges