$12^{2}_{316}$ - Minimal pinning sets
Pinning sets for 12^2_316 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_316 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 232
of which optimal: 1
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04417
on average over minimal pinning sets: 2.56667
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, 3, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 6, 7, 11, 12}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 6, 7, 10, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
c (minimal)
• {1, 2, 4, 6, 8, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
d (minimal)
• {1, 2, 6, 8, 11, 12}
6
[2, 2, 2, 3, 3, 3]
2.50
e (minimal)
• {1, 2, 6, 8, 10, 11}
6
[2, 2, 2, 3, 3, 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
1
0
0
2.4
6
0
5
7
2.64
7
0
0
42
2.86
8
0
0
70
3.02
9
0
0
64
3.14
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
5
226
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,2],[0,1,5,6],[0,7,7,0],[1,7,8,5],[1,4,8,2],[2,8,9,9],[3,9,4,3],[4,9,6,5],[6,8,7,6]]
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],[3,13,4,14],[8,4,9,5],[16,12,17,11],[2,18,3,19],[12,9,13,10],[17,10,18,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,2,-12,-3)(18,3,-19,-4)(8,5,-9,-6)(14,19,-15,-20)(1,16,-2,-17)(10,17,-11,-18)(6,7,-1,-8)(4,9,-5,-10)(15,12,-16,-13)(20,13,-7,-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,-17,10,-5,8)(-2,11,17)(-3,18,-11)(-4,-10,-18)(-6,-8)(-7,6,-9,4,-19,14)(-12,15,19,3)(-13,20,-15)(-14,-20)(-16,1,7,13)(2,16,12)(5,9)
Multiloop annotated with half-edges
12^2_316 annotated with half-edges