$12^{2}_{297}$ - Minimal pinning sets
Pinning sets for 12^2_297 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_297 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 276
of which optimal: 2
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10221
on average over minimal pinning sets: 2.70884
on average over optimal pinning sets: 2.6
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 7, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {2, 4, 6, 7, 11}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 3, 4, 7, 8, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 2, 4, 7, 8, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
c (minimal)
• {1, 3, 4, 6, 7, 11, 12}
7
[2, 2, 3, 3, 3, 3, 3]
2.71
d (minimal)
• {1, 3, 4, 6, 7, 9, 12}
7
[2, 2, 3, 3, 3, 3, 4]
2.86
e (minimal)
• {1, 2, 4, 6, 7, 9, 12}
7
[2, 2, 3, 3, 3, 3, 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
2
0
0
2.6
6
0
2
13
2.8
7
0
3
46
2.95
8
0
0
83
3.08
9
0
0
77
3.19
10
0
0
39
3.26
11
0
0
10
3.31
12
0
0
1
3.33
Total
2
5
269
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
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],[2,9,7,3],[3,6,9,3],[4,9,9,5],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,15,6,14],[3,13,4,14],[10,19,11,20],[1,16,2,15],[6,2,7,3],[9,12,10,13],[18,11,19,12],[16,8,17,7],[17,8,18,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(16,7,-17,-8)(17,10,-18,-11)(8,11,-9,-12)(1,12,-2,-13)(13,20,-14,-5)(14,3,-15,-4)(9,18,-10,-19)(2,19,-3,-20)(6,15,-7,-16)
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,-5)(-2,-20,13)(-3,14,20)(-4,5,-14)(-6,-16,-8,-12,1)(-7,16)(-9,-19,2,12)(-10,17,7,15,3,19)(-11,8,-17)(-15,6,4)(-18,9,11)(10,18)
Multiloop annotated with half-edges
12^2_297 annotated with half-edges