$12^{2}_{85}$ - Minimal pinning sets
Pinning sets for 12^2_85 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_85 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, 3, 6, 9, 10}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 6, 8, 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, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,3,4],[0,5,6,0],[0,4,1,1],[1,3,7,5],[2,4,8,8],[2,8,9,7],[4,6,9,9],[5,9,6,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[3,12,4,1],[2,20,3,13],[11,4,12,5],[1,14,2,13],[14,19,15,20],[5,15,6,16],[7,10,8,11],[8,18,9,19],[6,17,7,16],[17,9,18,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,2,-10,-3)(18,7,-19,-8)(5,8,-6,-9)(1,10,-2,-11)(3,16,-4,-17)(17,4,-18,-5)(6,19,-7,-20)(15,20,-16,-13)(12,13,-1,-14)(14,11,-15,-12)
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,-11,14)(-2,9,-6,-20,15,11)(-3,-17,-5,-9)(-4,17)(-7,18,4,16,20)(-8,5,-18)(-10,1,13,-16,3)(-12,-14)(-13,12,-15)(-19,6,8)(2,10)(7,19)
Multiloop annotated with half-edges
12^2_85 annotated with half-edges