$12^{1}_{73}$ - Minimal pinning sets
Pinning sets for 12^1_73 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_73 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, 3, 5, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 5, 6, 9}
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, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,6,7,8],[0,8,8,0],[1,8,7,1],[1,7,9,6],[2,5,9,9],[2,9,5,4],[2,4,3,3],[5,7,6,6]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[13,4,14,5],[19,8,20,9],[6,2,7,1],[3,12,4,13],[14,18,15,17],[9,17,10,16],[11,18,12,19],[7,2,8,3],[15,11,16,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,3,-15,-4)(5,10,-6,-11)(18,7,-19,-8)(11,4,-12,-5)(12,9,-13,-10)(6,13,-7,-14)(20,15,-1,-16)(16,1,-17,-2)(2,17,-3,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16)(-2,-18,-8,-20,-16)(-3,14,-7,18)(-4,11,-6,-14)(-5,-11)(-9,12,4,-15,20)(-10,5,-12)(-13,6,10)(-17,2)(-19,8)(1,15,3,17)(7,13,9,19)
Loop annotated with half-edges
12^1_73 annotated with half-edges