$12^{1}_{235}$ - Minimal pinning sets
Pinning sets for 12^1_235 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_235 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, 5, 6, 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: [[1,2,2,3],[0,4,4,5],[0,6,6,0],[0,7,7,4],[1,3,5,1],[1,4,8,6],[2,5,9,2],[3,9,8,3],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[8,15,9,16],[19,10,20,11],[1,6,2,7],[16,7,17,8],[17,14,18,15],[11,18,12,19],[5,2,6,3],[13,4,14,5],[12,4,13,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,9,-1,-10)(10,1,-11,-2)(18,3,-19,-4)(13,4,-14,-5)(16,7,-17,-8)(14,11,-15,-12)(5,12,-6,-13)(8,15,-9,-16)(6,17,-7,-18)(2,19,-3,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,-20,-10)(-3,18,-7,16,-9,20)(-4,13,-6,-18)(-5,-13)(-8,-16)(-11,14,4,-19,2)(-12,5,-14)(-15,8,-17,6,12)(1,9,15,11)(3,19)(7,17)
Loop annotated with half-edges
12^1_235 annotated with half-edges