$12^{2}_{21}$ - Minimal pinning sets
Pinning sets for 12^2_21 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_21 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 506
of which optimal: 6
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04228
on average over minimal pinning sets: 2.48571
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)
• {4, 6, 9, 10}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 4, 6, 9}
4
[2, 2, 2, 4]
2.50
C (optimal)
• {4, 6, 7, 9}
4
[2, 2, 2, 3]
2.25
D (optimal)
• {4, 6, 9, 11}
4
[2, 2, 2, 5]
2.75
E (optimal)
• {4, 6, 8, 9}
4
[2, 2, 2, 5]
2.75
F (optimal)
• {3, 4, 6, 9}
4
[2, 2, 2, 4]
2.50
a (minimal)
• {2, 4, 5, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
6
0
0
2.5
5
0
1
33
2.72
6
0
0
84
2.89
7
0
0
126
3.02
8
0
0
126
3.11
9
0
0
84
3.19
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
6
1
499
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,7],[0,8,9,5],[0,5,1,1],[1,4,3,2],[2,9,7,7],[2,6,6,8],[3,7,9,9],[3,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[3,16,4,1],[2,9,3,10],[12,15,13,16],[4,7,5,8],[1,11,2,10],[11,8,12,9],[14,20,15,17],[13,20,14,19],[6,18,7,19],[5,18,6,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (2,5,-3,-6)(9,6,-10,-7)(16,7,-1,-8)(8,15,-9,-16)(10,13,-11,-14)(1,14,-2,-15)(20,3,-17,-4)(4,17,-5,-18)(18,11,-19,-12)(12,19,-13,-20)
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,-15,8)(-2,-6,9,15)(-3,20,-13,10,6)(-4,-18,-12,-20)(-5,2,14,-11,18)(-7,16,-9)(-8,-16)(-10,-14,1,7)(-17,4)(-19,12)(3,5,17)(11,13,19)
Multiloop annotated with half-edges
12^2_21 annotated with half-edges