$12^{2}_{192}$ - Minimal pinning sets
Pinning sets for 12^2_192 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_192 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, 8, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 8, 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, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,7],[1,7,7,8],[2,8,8,6],[3,5,9,9],[3,9,4,4],[4,9,5,5],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[16,9,1,10],[10,15,11,16],[11,8,12,9],[1,12,2,13],[5,14,6,15],[7,20,8,17],[2,20,3,19],[13,4,14,5],[6,18,7,17],[3,18,4,19]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(8,5,-9,-6)(13,6,-14,-7)(14,9,-15,-10)(2,11,-3,-12)(7,12,-8,-13)(20,3,-17,-4)(16,17,-1,-18)(18,15,-19,-16)(4,19,-5,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-15,18)(-2,-12,7,-14,-10)(-3,20,-5,8,12)(-4,-20)(-6,13,-8)(-7,-13)(-9,14,6)(-11,2)(-16,-18)(-17,16,-19,4)(1,17,3,11)(5,19,15,9)
Multiloop annotated with half-edges
12^2_192 annotated with half-edges