$12^{2}_{209}$ - Minimal pinning sets
Pinning sets for 12^2_209 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_209 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, 5, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {2, 3, 5, 7, 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: [[1,2,3,4],[0,5,5,2],[0,1,6,6],[0,7,8,9],[0,9,9,5],[1,4,6,1],[2,5,7,2],[3,6,8,8],[3,7,7,9],[3,8,4,4]]
PD code (use to draw this multiloop with SnapPy): [[16,7,1,8],[8,17,9,20],[15,19,16,20],[6,3,7,4],[1,11,2,10],[17,10,18,9],[18,14,19,15],[4,14,5,13],[5,12,6,13],[2,11,3,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (4,1,-5,-2)(14,5,-15,-6)(7,18,-8,-19)(8,15,-9,-16)(16,9,-1,-10)(3,10,-4,-11)(11,2,-12,-3)(19,12,-20,-13)(13,20,-14,-17)(17,6,-18,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,10)(-2,11,-4)(-3,-11)(-5,14,20,12,2)(-6,17,-14)(-7,-19,-13,-17)(-8,-16,-10,3,-12,19)(-9,16)(-15,8,18,6)(-18,7)(-20,13)(1,9,15,5)
Multiloop annotated with half-edges
12^2_209 annotated with half-edges