$12^{2}_{187}$ - Minimal pinning sets
Pinning sets for 12^2_187 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_187 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, 5, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 4, 9, 11}
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],[0,4,2,0],[0,1,5,3],[0,2,6,4],[1,3,7,5],[2,4,8,8],[3,9,7,7],[4,6,6,9],[5,9,9,5],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,5,8,6],[8,19,9,20],[1,9,2,10],[10,4,11,5],[11,18,12,19],[2,14,3,15],[15,3,16,4],[17,12,18,13],[13,16,14,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(17,2,-18,-3)(14,3,-15,-4)(5,16,-6,-17)(10,19,-11,-20)(6,7,-1,-8)(20,9,-7,-10)(18,11,-19,-12)(15,12,-16,-13)(4,13,-5,-14)
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)(-2,17,-6,-8)(-3,14,-5,-17)(-4,-14)(-7,6,16,12,-19,10)(-9,20,-11,18,2)(-10,-20)(-12,15,3,-18)(-13,4,-15)(-16,5,13)(1,7,9)(11,19)
Multiloop annotated with half-edges
12^2_187 annotated with half-edges