$12^{2}_{83}$ - Minimal pinning sets
Pinning sets for 12^2_83
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_83
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, 4, 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, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,6,6,7],[0,7,4,0],[1,3,5,1],[1,4,8,6],[2,5,9,2],[2,9,8,3],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[16,20,1,17],[17,12,18,11],[15,6,16,7],[19,1,20,2],[12,19,13,18],[13,10,14,11],[7,14,8,15],[5,2,6,3],[9,4,10,5],[8,4,9,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (3,16,-4,-1)(1,6,-2,-7)(7,2,-8,-3)(13,4,-14,-5)(11,8,-12,-9)(9,20,-10,-17)(5,12,-6,-13)(18,15,-19,-16)(17,10,-18,-11)(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,-7,-3)(-2,7)(-4,13,-6,1)(-5,-13)(-8,11,-18,-16,3)(-9,-17,-11)(-10,17)(-12,5,-14,-20,9)(-15,18,10,20)(-19,14,4,16)(2,6,12,8)(15,19)
Multiloop annotated with half-edges
12^2_83 annotated with half-edges