$12^{4}_{10}$ - Minimal pinning sets
Pinning sets for 12^4_10 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^4_10 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 372
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: 3.10755
on average over minimal pinning sets: 2.72917
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 6, 9}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 2, 4, 5, 6, 9}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 4, 5, 6, 8, 9}
6
[2, 2, 3, 3, 3, 4]
2.83
c (minimal)
• {1, 4, 5, 6, 9, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
d (minimal)
• {1, 4, 5, 6, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
e (minimal)
• {1, 3, 4, 6, 8, 9}
6
[2, 2, 3, 3, 3, 4]
2.83
f (minimal)
• {1, 3, 4, 6, 9, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
g (minimal)
• {1, 3, 4, 6, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.5
5
0
0
8
2.75
6
0
7
28
2.89
7
0
0
82
3.02
8
0
0
109
3.12
9
0
0
86
3.2
10
0
0
40
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
7
364
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,6],[0,6,7,7],[0,8,5,1],[1,4,8,2],[2,9,9,3],[3,9,8,3],[4,7,9,5],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[4,10,1,5],[5,11,6,16],[3,15,4,16],[9,20,10,17],[1,12,2,11],[6,2,7,3],[14,17,15,18],[19,8,20,9],[12,8,13,7],[18,13,19,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(6,13,-7,-14)(20,7,-17,-8)(2,9,-3,-10)(19,14,-20,-15)(1,16,-2,-11)(11,10,-12,-5)(12,3,-13,-4)(8,17,-9,-18)(15,18,-16,-19)
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,-5)(-2,-10,11)(-3,12,10)(-4,5,-12)(-6,-14,19,-16,1)(-7,20,14)(-8,-18,15,-20)(-9,2,16,18)(-13,6,4)(-15,-19)(-17,8)(3,9,17,7,13)
Multiloop annotated with half-edges
12^4_10 annotated with half-edges