$12^{2}_{137}$ - Minimal pinning sets
Pinning sets for 12^2_137 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_137 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, 6, 7}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 5, 6, 7}
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, 4, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,3,0],[0,2,7,1],[1,7,8,5],[1,4,9,9],[2,8,7,7],[3,6,6,4],[4,6,9,9],[5,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[14,5,1,6],[6,12,7,11],[4,13,5,14],[1,13,2,12],[7,15,8,20],[10,19,11,20],[3,16,4,17],[2,16,3,15],[8,17,9,18],[18,9,19,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,2,-12,-3)(17,6,-18,-7)(8,3,-9,-4)(9,12,-10,-13)(1,10,-2,-11)(5,16,-6,-17)(7,18,-8,-19)(19,4,-20,-5)(20,13,-15,-14)(14,15,-1,-16)
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,-3,8,18,6,16)(-2,11)(-4,19,-8)(-5,-17,-7,-19)(-6,17)(-9,-13,20,4)(-10,1,15,13)(-12,9,3)(-14,-16,5,-20)(-15,14)(-18,7)(2,10,12)
Multiloop annotated with half-edges
12^2_137 annotated with half-edges