$12^{1}_{156}$ - Minimal pinning sets
Pinning sets for 12^1_156 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_156 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, 4, 6, 7}
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, 4, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,2],[0,3,4,0],[0,4,5,0],[1,5,6,6],[1,7,5,2],[2,4,8,3],[3,8,7,3],[4,6,9,9],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[19,10,20,11],[12,2,13,1],[18,7,19,8],[9,2,10,3],[13,9,14,8],[6,17,7,18],[3,17,4,16],[14,5,15,6],[4,15,5,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,2,-12,-3)(15,4,-16,-5)(5,10,-6,-11)(16,7,-17,-8)(1,12,-2,-13)(13,20,-14,-1)(3,14,-4,-15)(8,17,-9,-18)(18,9,-19,-10)(6,19,-7,-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,-13)(-2,11,-6,-20,13)(-3,-15,-5,-11)(-4,15)(-7,16,4,14,20)(-8,-18,-10,5,-16)(-9,18)(-12,1,-14,3)(-17,8)(-19,6,10)(2,12)(7,19,9,17)
Loop annotated with half-edges
12^1_156 annotated with half-edges