$12^{2}_{16}$ - Minimal pinning sets
Pinning sets for 12^2_16 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_16 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 256
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96564
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 8, 11}
4
[2, 2, 2, 2]
2.00
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
2.0
5
0
0
8
2.4
6
0
0
28
2.67
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
1
0
255
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,7],[0,7,6,8],[0,8,1,1],[1,8,8,2],[2,3,9,9],[2,9,9,3],[3,5,5,4],[6,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[3,16,4,1],[2,20,3,17],[15,8,16,9],[4,12,5,11],[1,18,2,17],[19,9,20,10],[5,14,6,15],[7,12,8,13],[10,18,11,19],[13,6,14,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,4,-12,-5)(13,6,-14,-7)(20,7,-17,-8)(9,2,-10,-3)(5,10,-6,-11)(3,12,-4,-13)(1,14,-2,-15)(16,17,-1,-18)(18,15,-19,-16)(8,19,-9,-20)
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,-15,18)(-2,9,19,15)(-3,-13,-7,20,-9)(-4,11,-6,13)(-5,-11)(-8,-20)(-10,5,-12,3)(-14,1,17,7)(-16,-18)(-17,16,-19,8)(2,14,6,10)(4,12)
Multiloop annotated with half-edges
12^2_16 annotated with half-edges