$12^{1}_{159}$ - Minimal pinning sets
Pinning sets for 12^1_159 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_159 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 128
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: 2.90623
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, 3, 4, 9}
5
[2, 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
5
1
0
0
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,7,7],[0,8,8,6],[0,9,5,5],[1,4,4,9],[1,9,9,3],[2,8,8,2],[3,7,7,3],[4,6,6,5]]
PD code (use to draw this loop with SnapPy): [[15,20,16,1],[3,14,4,15],[4,19,5,20],[16,7,17,8],[1,10,2,11],[11,2,12,3],[13,8,14,9],[18,5,19,6],[6,17,7,18],[9,12,10,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,2,-16,-3)(13,4,-14,-5)(10,5,-11,-6)(18,7,-19,-8)(6,9,-7,-10)(20,11,-1,-12)(3,14,-4,-15)(1,16,-2,-17)(12,17,-13,-18)(8,19,-9,-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,-17,12)(-2,15,-4,13,17)(-3,-15)(-5,10,-7,18,-13)(-6,-10)(-8,-20,-12,-18)(-9,6,-11,20)(-14,3,-16,1,11,5)(-19,8)(2,16)(4,14)(7,9,19)
Loop annotated with half-edges
12^1_159 annotated with half-edges