$12^{2}_{110}$ - Minimal pinning sets
Pinning sets for 12^2_110 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_110 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 132
of which optimal: 2
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98691
on average over minimal pinning sets: 2.42381
on average over optimal pinning sets: 2.41667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 6, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 5, 6, 7, 12}
6
[2, 2, 2, 2, 3, 4]
2.50
a (minimal)
• {1, 2, 4, 5, 6, 7, 9}
7
[2, 2, 2, 2, 3, 3, 3]
2.43
b (minimal)
• {1, 2, 5, 6, 7, 9, 11}
7
[2, 2, 2, 2, 3, 3, 3]
2.43
c (minimal)
• {1, 2, 3, 4, 5, 6, 7}
7
[2, 2, 2, 2, 3, 3, 3]
2.43
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.42
7
0
3
12
2.63
8
0
0
37
2.87
9
0
0
43
3.06
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
3
127
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 8]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,6,0],[0,7,8,1],[1,8,9,5],[1,4,9,9],[2,7,7,2],[3,6,6,8],[3,7,9,4],[4,8,5,5]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,6,12,5],[19,9,20,10],[1,7,2,6],[12,16,13,15],[4,14,5,15],[8,18,9,19],[7,18,8,17],[2,17,3,16],[13,3,14,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,10,-12,-1)(2,19,-3,-20)(3,8,-4,-9)(17,6,-18,-7)(1,12,-2,-13)(13,20,-14,-11)(14,9,-15,-10)(15,4,-16,-5)(5,16,-6,-17)(7,18,-8,-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,-13,-11)(-2,-20,13)(-3,-9,14,20)(-4,15,9)(-5,-17,-7,-19,2,12,10,-15)(-6,17)(-8,3,19)(-10,11,-14)(-12,1)(-16,5)(-18,7)(4,8,18,6,16)
Multiloop annotated with half-edges
12^2_110 annotated with half-edges