$12^{2}_{196}$ - Minimal pinning sets
Pinning sets for 12^2_196 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_196 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 220
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04183
on average over minimal pinning sets: 2.55619
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 5, 6, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 4, 5, 7, 12}
6
[2, 2, 2, 3, 3, 4]
2.67
c (minimal)
• {1, 2, 5, 6, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
d (minimal)
• {1, 2, 4, 5, 6, 7, 10}
7
[2, 2, 2, 3, 3, 3, 4]
2.71
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.4
6
0
3
7
2.63
7
0
1
36
2.83
8
0
0
66
3.01
9
0
0
63
3.14
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
4
215
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,7,3],[0,2,7,7],[0,8,8,5],[1,4,9,6],[1,5,9,7],[2,6,3,3],[4,9,9,4],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,5,12,6],[6,9,7,10],[7,19,8,20],[1,15,2,16],[16,4,17,5],[12,17,13,18],[18,8,19,9],[14,2,15,3],[3,13,4,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(19,2,-20,-3)(16,5,-17,-6)(3,6,-4,-7)(12,9,-13,-10)(10,11,-1,-12)(8,13,-9,-14)(20,15,-11,-16)(4,17,-5,-18)(7,18,-8,-19)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-9,12)(-2,19,-8,-14)(-3,-7,-19)(-4,-18,7)(-5,16,-11,10,-13,8,18)(-6,3,-20,-16)(-10,-12)(-15,20,2)(-17,4,6)(1,11,15)(5,17)(9,13)
Multiloop annotated with half-edges
12^2_196 annotated with half-edges