$12^{1}_{78}$ - Minimal pinning sets
Pinning sets for 12^1_78 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_78 Pinning data
Pinning number of this loop: 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, 5, 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 loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,5],[0,5,6,1],[2,6,7,7],[2,7,8,3],[3,9,9,4],[4,8,5,4],[5,7,9,9],[6,8,8,6]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[19,12,20,13],[14,12,15,11],[1,18,2,19],[15,7,16,6],[17,10,18,11],[2,8,3,7],[16,5,17,6],[9,4,10,5],[8,4,9,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,5,-1,-6)(6,1,-7,-2)(13,2,-14,-3)(3,12,-4,-13)(4,19,-5,-20)(14,7,-15,-8)(8,11,-9,-12)(16,9,-17,-10)(18,15,-19,-16)(10,17,-11,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6)(-2,13,-4,-20,-6)(-3,-13)(-5,20)(-7,14,2)(-8,-12,3,-14)(-9,16,-19,4,12)(-10,-18,-16)(-11,8,-15,18)(-17,10)(1,5,19,15,7)(9,11,17)
Loop annotated with half-edges
12^1_78 annotated with half-edges