$12^{2}_{248}$ - Minimal pinning sets
Pinning sets for 12^2_248 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_248 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 96
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: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 6, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 4, 6, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,5,5],[0,6,7,0],[1,7,8,1],[2,8,8,2],[3,9,9,7],[3,6,9,4],[4,9,5,5],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,8,14,7],[11,6,12,7],[19,1,20,2],[8,15,9,14],[5,10,6,11],[2,17,3,16],[18,15,19,16],[9,4,10,5],[17,4,18,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,14,-2,-15)(7,2,-8,-3)(9,4,-10,-5)(5,18,-6,-19)(3,8,-4,-9)(16,11,-17,-12)(19,6,-20,-7)(15,20,-16,-13)(13,12,-14,-1)(10,17,-11,-18)
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,-15,-13)(-2,7,-20,15)(-3,-9,-5,-19,-7)(-4,9)(-6,19)(-8,3)(-10,-18,5)(-11,16,20,6,18)(-12,13,-16)(-14,1)(-17,10,4,8,2,14,12)(11,17)
Multiloop annotated with half-edges
12^2_248 annotated with half-edges