$12^{2}_{143}$ - Minimal pinning sets
Pinning sets for 12^2_143 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_143 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96906
on average over minimal pinning sets: 2.2
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)
• {2, 3, 4, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
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
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
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
2
0
190
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: [[1,2,3,3],[0,3,4,5],[0,6,6,7],[0,7,1,0],[1,8,9,5],[1,4,9,9],[2,9,8,2],[2,8,8,3],[4,7,7,6],[4,6,5,5]]
PD code (use to draw this multiloop with SnapPy): [[9,16,10,1],[11,8,12,9],[15,20,16,17],[10,2,11,1],[4,7,5,8],[12,5,13,6],[17,14,18,15],[19,2,20,3],[3,18,4,19],[6,13,7,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(16,3,-1,-4)(7,4,-8,-5)(2,9,-3,-10)(17,10,-18,-11)(5,12,-6,-13)(13,6,-14,-7)(19,14,-20,-15)(11,20,-12,-17)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,4)(-2,-10,17,-12,5,-8)(-3,16,18,10)(-4,7,-14,19,-16)(-5,-13,-7)(-6,13)(-9,2)(-11,-17)(-15,-19)(-18,15,-20,11)(1,3,9)(6,12,20,14)
Multiloop annotated with half-edges
12^2_143 annotated with half-edges