$12^{2}_{185}$ - Minimal pinning sets
Pinning sets for 12^2_185 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_185 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)
• {1, 2, 4, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 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: [[1,1,2,3],[0,4,5,0],[0,5,6,7],[0,7,8,8],[1,9,9,6],[1,6,6,2],[2,5,5,4],[2,9,8,3],[3,7,9,3],[4,8,7,4]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,5,8,6],[19,16,20,17],[1,13,2,14],[4,10,5,11],[8,18,9,17],[9,18,10,19],[12,15,13,16],[2,15,3,14],[11,3,12,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(17,2,-18,-3)(4,9,-5,-10)(10,5,-11,-6)(8,11,-9,-12)(15,12,-16,-13)(13,18,-14,-19)(19,14,-20,-15)(3,16,-4,-17)(1,20,-2,-7)
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,-7)(-2,17,-4,-10,-6,7)(-3,-17)(-5,10)(-8,-12,15,-20,1)(-9,4,16,12)(-11,8,6)(-13,-19,-15)(-14,19)(-16,3,-18,13)(2,20,14,18)(5,9,11)
Multiloop annotated with half-edges
12^2_185 annotated with half-edges