$12^{2}_{39}$ - Minimal pinning sets
Pinning sets for 12^2_39 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_39 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 64
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.85421
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 8, 9}
6
[2, 2, 2, 2, 2, 2]
2.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
1
0
0
2.0
7
0
0
6
2.38
8
0
0
15
2.67
9
0
0
20
2.89
10
0
0
15
3.07
11
0
0
6
3.21
12
0
0
1
3.33
Total
1
0
63
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 4, 4, 5, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,2],[0,3,4,0],[0,5,6,0],[1,6,6,7],[1,7,5,5],[2,4,4,8],[2,8,3,3],[3,9,9,4],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[5,10,6,1],[9,4,10,5],[6,2,7,1],[8,20,9,11],[15,3,16,4],[2,16,3,17],[7,12,8,11],[14,19,15,20],[17,13,18,12],[18,13,19,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-11,-2)(7,2,-8,-3)(3,8,-4,-9)(9,4,-10,-5)(17,12,-18,-13)(13,18,-14,-19)(10,11,-1,-12)(19,14,-20,-15)(6,15,-7,-16)(16,5,-17,-6)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,14,18,12)(-2,7,15,-20)(-3,-9,-5,16,-7)(-4,9)(-6,-16)(-8,3)(-10,-12,17,5)(-11,10,4,8,2)(-13,-19,-15,6,-17)(-14,19)(-18,13)(1,11)
Multiloop annotated with half-edges
12^2_39 annotated with half-edges