$12^{2}_{18}$ - Minimal pinning sets
Pinning sets for 12^2_18 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_18 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 320
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03463
on average over minimal pinning sets: 2.325
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 3, 5, 9}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
1
8
2.56
6
0
0
34
2.77
7
0
0
71
2.94
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
1
318
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,6,6,3],[0,2,1,0],[1,6,7,8],[1,8,9,6],[2,5,4,2],[4,9,9,8],[4,7,9,5],[5,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,19,14,18],[11,2,12,3],[19,1,20,2],[14,4,15,5],[9,17,10,18],[3,10,4,11],[15,7,16,8],[5,8,6,9],[6,16,7,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,12,-10,-1)(13,2,-14,-3)(14,5,-15,-6)(3,6,-4,-7)(1,8,-2,-9)(18,11,-19,-12)(4,15,-5,-16)(7,16,-8,-17)(17,20,-18,-13)(10,19,-11,-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,-9)(-2,13,-18,-12,9)(-3,-7,-17,-13)(-4,-16,7)(-5,14,2,8,16)(-6,3,-14)(-8,1,-10,-20,17)(-11,18,20)(-15,4,6)(-19,10,12)(5,15)(11,19)
Multiloop annotated with half-edges
12^2_18 annotated with half-edges