$12^{2}_{339}$ - Minimal pinning sets
Pinning sets for 12^2_339 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_339 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 216
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: 2.98842
on average over minimal pinning sets: 2.425
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, 3, 6, 7, 9}
5
[2, 2, 2, 2, 4]
2.40
a (minimal)
• {2, 3, 5, 6, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 2, 3, 5, 6, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
c (minimal)
• {1, 2, 3, 6, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
d (minimal)
• {2, 3, 4, 6, 8, 9}
6
[2, 2, 2, 2, 3, 5]
2.67
e (minimal)
• {2, 3, 5, 6, 9, 10}
6
[2, 2, 2, 2, 3, 5]
2.67
f (minimal)
• {2, 3, 5, 6, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
g (minimal)
• {2, 3, 6, 8, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
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
2.55
7
0
0
44
2.81
8
0
0
65
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
7
208
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,6,3],[0,2,7,7],[0,8,8,1],[1,9,9,6],[1,5,2,2],[3,9,8,3],[4,7,9,4],[5,8,7,5]]
PD code (use to draw this multiloop with SnapPy): [[14,5,1,6],[6,11,7,12],[13,20,14,15],[4,19,5,20],[1,10,2,11],[7,17,8,16],[12,16,13,15],[18,3,19,4],[9,2,10,3],[17,9,18,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,14,-8,-1)(20,1,-15,-2)(12,3,-13,-4)(6,19,-7,-20)(13,8,-14,-9)(4,9,-5,-10)(18,5,-19,-6)(2,15,-3,-16)(11,16,-12,-17)(17,10,-18,-11)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,-7)(-2,-16,11,-18,-6,-20)(-3,12,16)(-4,-10,17,-12)(-5,18,10)(-8,13,3,15,1)(-9,4,-13)(-11,-17)(-14,7,19,5,9)(-15,2)(-19,6)(8,14)
Multiloop annotated with half-edges
12^2_339 annotated with half-edges