$12^{2}_{82}$ - Minimal pinning sets
Pinning sets for 12^2_82 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_82 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, 7}
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: [[1,1,2,3],[0,4,2,0],[0,1,5,6],[0,6,4,4],[1,3,3,7],[2,7,8,8],[2,9,7,3],[4,6,9,5],[5,9,9,5],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,3,6,4],[6,19,7,20],[1,13,2,14],[14,2,15,3],[9,18,10,19],[7,12,8,13],[15,8,16,9],[17,10,18,11],[11,16,12,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(12,1,-13,-2)(6,15,-7,-16)(8,17,-9,-18)(14,19,-15,-20)(3,20,-4,-5)(18,7,-19,-8)(16,9,-17,-10)(13,10,-14,-11)(2,11,-3,-12)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-3,-5)(-2,-12)(-4,5)(-6,-16,-10,13,1)(-7,18,-9,16)(-8,-18)(-11,2,-13)(-14,-20,3,11)(-15,6,4,20)(-17,8,-19,14,10)(7,15,19)(9,17)
Multiloop annotated with half-edges
12^2_82 annotated with half-edges