$12^{2}_{35}$ - Minimal pinning sets
Pinning sets for 12^2_35 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_35 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 242
of which optimal: 3
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9801
on average over minimal pinning sets: 2.49524
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 6, 7}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 5, 7, 9}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 3, 5, 7}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 3, 5, 7, 8, 10}
6
[2, 2, 2, 2, 4, 4]
2.67
b (minimal)
• {1, 3, 5, 7, 10, 11}
6
[2, 2, 2, 2, 4, 4]
2.67
c (minimal)
• {1, 3, 4, 5, 7, 8}
6
[2, 2, 2, 2, 4, 4]
2.67
d (minimal)
• {1, 3, 4, 5, 7, 11}
6
[2, 2, 2, 2, 4, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.27
6
0
4
18
2.61
7
0
0
54
2.85
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
3
4
235
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,3,3],[0,2,2,4],[1,3,6,7],[1,7,8,2],[4,8,8,9],[4,9,9,5],[5,9,6,6],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[10,3,1,4],[4,9,5,10],[2,20,3,11],[1,20,2,19],[8,18,9,19],[5,12,6,11],[15,7,16,8],[17,12,18,13],[6,14,7,15],[16,14,17,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(16,5,-17,-6)(2,7,-3,-8)(14,17,-15,-18)(4,15,-5,-16)(6,19,-7,-20)(20,9,-11,-10)(10,11,-1,-12)(12,3,-13,-4)(18,13,-19,-14)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,-3,12)(-2,-8)(-4,-16,-6,-20,-10,-12)(-5,16)(-7,2,-9,20)(-11,10)(-13,18,-15,4)(-14,-18)(-17,14,-19,6)(1,11,9)(3,7,19,13)(5,15,17)
Multiloop annotated with half-edges
12^2_35 annotated with half-edges