$12^{2}_{193}$ - Minimal pinning sets
Pinning sets for 12^2_193 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_193 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, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {2, 4, 5, 6, 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, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,6,3],[0,2,7,7],[0,8,9,9],[1,9,8,1],[1,7,2,2],[3,6,8,3],[4,7,5,9],[4,8,5,4]]
PD code (use to draw this multiloop with SnapPy): [[5,16,6,1],[13,4,14,5],[15,20,16,17],[6,20,7,19],[1,11,2,10],[3,12,4,13],[14,18,15,17],[7,18,8,19],[11,8,12,9],[2,9,3,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,2,-12,-3)(9,6,-10,-7)(16,7,-1,-8)(8,15,-9,-16)(1,10,-2,-11)(3,14,-4,-15)(5,20,-6,-17)(17,4,-18,-5)(13,18,-14,-19)(19,12,-20,-13)
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,-15,8)(-2,11)(-4,17,-6,9,15)(-5,-17)(-7,16,-9)(-8,-16)(-10,1,7)(-12,19,-14,3)(-13,-19)(-18,13,-20,5)(2,10,6,20,12)(4,14,18)
Multiloop annotated with half-edges
12^2_193 annotated with half-edges