$12^{2}_{58}$ - Minimal pinning sets
Pinning sets for 12^2_58 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_58 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 400
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10335
on average over minimal pinning sets: 2.59167
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 11}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 3, 5, 6, 11}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 2, 4, 5, 12}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {1, 2, 3, 5, 6, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
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
2.5
5
0
2
8
2.72
6
0
1
40
2.89
7
0
0
90
3.02
8
0
0
116
3.12
9
0
0
90
3.21
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
3
396
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,7,3,3],[0,2,2,4],[0,3,7,1],[1,7,8,6],[1,5,8,9],[2,9,5,4],[5,9,9,6],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[7,10,8,1],[6,20,7,11],[9,13,10,14],[8,13,9,12],[1,12,2,11],[15,5,16,6],[16,19,17,20],[14,3,15,2],[18,4,19,5],[17,4,18,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,11,-1,-12)(12,1,-13,-2)(19,2,-20,-3)(4,7,-5,-8)(9,18,-10,-19)(16,5,-17,-6)(6,17,-7,-18)(20,13,-11,-14)(3,14,-4,-15)(15,8,-16,-9)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12)(-2,19,-10,-12)(-3,-15,-9,-19)(-4,-8,15)(-5,16,8)(-6,-18,9,-16)(-7,4,14,-11,10,18)(-13,20,2)(-14,3,-20)(-17,6)(1,11,13)(5,7,17)
Multiloop annotated with half-edges
12^2_58 annotated with half-edges