$12^{1}_{46}$ - Minimal pinning sets
Pinning sets for 12^1_46 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_46 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 160
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97043
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 9}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 4, 6, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
1
7
2.5
7
0
0
26
2.74
8
0
0
45
2.92
9
0
0
45
3.07
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
158
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,7,0],[0,4,4,1],[1,3,3,5],[1,4,7,8],[2,8,8,9],[2,9,9,5],[5,9,6,6],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,11,9,12],[6,19,7,20],[1,10,2,11],[9,2,10,3],[12,3,13,4],[16,5,17,6],[18,13,19,14],[4,15,5,16],[17,15,18,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(15,4,-16,-5)(2,5,-3,-6)(12,7,-13,-8)(8,11,-9,-12)(18,9,-19,-10)(20,13,-1,-14)(3,16,-4,-17)(6,17,-7,-18)(10,19,-11,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14)(-2,-6,-18,-10,-20,-14)(-3,-17,6)(-4,15,1,13,7,17)(-5,2,-15)(-7,12,-9,18)(-8,-12)(-11,8,-13,20)(-16,3,5)(-19,10)(4,16)(9,11,19)
Loop annotated with half-edges
12^1_46 annotated with half-edges