$12^{2}_{189}$ - Minimal pinning sets
Pinning sets for 12^2_189 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_189 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 184
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97614
on average over minimal pinning sets: 2.38333
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 5, 6, 8, 9}
6
[2, 2, 2, 2, 3, 4]
2.50
b (minimal)
• {1, 2, 5, 6, 9, 12}
6
[2, 2, 2, 2, 3, 4]
2.50
c (minimal)
• {1, 2, 5, 6, 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
2.2
6
0
3
7
2.5
7
0
0
33
2.76
8
0
0
54
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
3
180
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,7,7],[0,8,8,5],[1,4,2,1],[2,9,9,3],[3,9,8,3],[4,7,9,4],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[16,5,1,6],[6,14,7,13],[15,12,16,13],[4,20,5,17],[1,9,2,8],[14,8,15,7],[11,17,12,18],[19,3,20,4],[9,3,10,2],[18,10,19,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,7,-1,-8)(20,1,-17,-2)(14,3,-15,-4)(5,12,-6,-13)(6,15,-7,-16)(19,8,-20,-9)(13,10,-14,-11)(11,4,-12,-5)(2,17,-3,-18)(9,18,-10,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,8)(-2,-18,9,-20)(-3,14,10,18)(-4,11,-14)(-5,-13,-11)(-6,-16,-8,19,-10,13)(-7,16)(-9,-19)(-12,5)(-15,6,12,4)(-17,2)(1,7,15,3,17)
Multiloop annotated with half-edges
12^2_189 annotated with half-edges