$12^{1}_{38}$ - Minimal pinning sets
Pinning sets for 12^1_38 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_38 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 192
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96906
on average over minimal pinning sets: 2.2
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, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 4, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,6],[0,6,7,7],[0,8,4,0],[1,3,5,5],[1,4,4,6],[1,5,9,2],[2,9,8,2],[3,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[8,5,9,6],[14,19,15,20],[10,2,11,1],[6,11,7,12],[12,7,13,8],[13,4,14,5],[18,15,19,16],[2,18,3,17],[3,16,4,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(5,2,-6,-3)(9,4,-10,-5)(3,8,-4,-9)(15,12,-16,-13)(20,13,-1,-14)(14,19,-15,-20)(11,16,-12,-17)(6,17,-7,-18)(18,7,-19,-8)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,4,8,-19,14)(-2,5,-10)(-3,-9,-5)(-4,9)(-6,-18,-8,3)(-7,18)(-11,-17,6,2)(-12,15,19,7,17)(-13,20,-15)(-14,-20)(-16,11,1,13)(12,16)
Loop annotated with half-edges
12^1_38 annotated with half-edges