$12^{2}_{341}$ - Minimal pinning sets
Pinning sets for 12^2_341 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_341 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 252
of which optimal: 6
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97224
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 6, 9, 10}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 3, 6, 9}
5
[2, 2, 2, 2, 4]
2.40
D (optimal)
• {1, 3, 4, 6, 9}
5
[2, 2, 2, 2, 4]
2.40
E (optimal)
• {1, 3, 6, 7, 9}
5
[2, 2, 2, 2, 4]
2.40
F (optimal)
• {1, 3, 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
6
0
0
2.33
6
0
0
27
2.65
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
6
0
246
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,5,6],[0,6,7,0],[1,7,5,1],[2,4,8,8],[2,8,9,3],[3,9,9,4],[5,9,6,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,14,8,15],[15,5,16,6],[19,1,20,2],[13,8,14,9],[4,12,5,13],[16,3,17,2],[18,9,19,10],[11,3,12,4],[17,11,18,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(14,3,-15,-4)(4,11,-5,-12)(20,5,-7,-6)(13,18,-14,-19)(19,12,-20,-13)(6,7,-1,-8)(16,9,-17,-10)(2,15,-3,-16)(8,17,-9,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-17,8)(-2,-16,-10)(-3,14,18,-9,16)(-4,-12,19,-14)(-5,20,12)(-6,-8,-18,13,-20)(-7,6)(-11,4,-15,2)(-13,-19)(1,7,5,11)(3,15)(9,17)
Multiloop annotated with half-edges
12^2_341 annotated with half-edges