$12^{2}_{84}$ - Minimal pinning sets
Pinning sets for 12^2_84 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_84 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, 6, 9}
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, 3, 4, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,5,6,7],[0,7,4,0],[1,3,5,1],[1,4,8,2],[2,8,8,9],[2,9,9,3],[5,9,6,6],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[16,20,1,17],[17,11,18,12],[15,8,16,9],[19,1,20,2],[10,18,11,19],[12,10,13,9],[5,14,6,15],[7,2,8,3],[13,4,14,5],[6,4,7,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,16,-6,-1)(1,4,-2,-5)(11,2,-12,-3)(13,6,-14,-7)(7,10,-8,-11)(20,9,-17,-10)(3,12,-4,-13)(18,15,-19,-16)(8,17,-9,-18)(14,19,-15,-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,-5)(-2,11,-8,-18,-16,5)(-3,-13,-7,-11)(-4,1,-6,13)(-9,20,-15,18)(-10,7,-14,-20)(-12,3)(-17,8,10)(-19,14,6,16)(2,4,12)(9,17)(15,19)
Multiloop annotated with half-edges
12^2_84 annotated with half-edges