$12^{1}_{146}$ - Minimal pinning sets
Pinning sets for 12^1_146 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_146 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 176
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97631
on average over minimal pinning sets: 2.34444
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, 3, 4, 5, 7}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 5, 7, 12}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 2, 3, 5, 7, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
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
2
7
2.5
7
0
0
30
2.75
8
0
0
51
2.94
9
0
0
49
3.08
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
2
173
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,4],[0,5,5,6],[0,6,1,0],[1,7,8,1],[2,9,6,2],[2,5,7,3],[4,6,9,8],[4,7,9,9],[5,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,13,1,14],[14,12,15,11],[4,19,5,20],[12,1,13,2],[15,10,16,11],[18,3,19,4],[5,3,6,2],[6,9,7,10],[16,7,17,8],[8,17,9,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,20,-16,-1)(13,4,-14,-5)(3,6,-4,-7)(12,7,-13,-8)(8,11,-9,-12)(17,10,-18,-11)(5,14,-6,-15)(1,16,-2,-17)(9,18,-10,-19)(19,2,-20,-3)
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,-17,-11,8,-13,-5,-15)(-2,19,-10,17)(-3,-7,12,-9,-19)(-4,13,7)(-6,3,-20,15)(-8,-12)(-14,5)(-16,1)(-18,9,11)(2,16,20)(4,6,14)(10,18)
Loop annotated with half-edges
12^1_146 annotated with half-edges