$12^{1}_{70}$ - Minimal pinning sets
Pinning sets for 12^1_70 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_70 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, 6, 7}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 5, 6, 7}
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: [[1,1,2,2],[0,3,4,0],[0,4,3,0],[1,2,5,6],[1,7,5,2],[3,4,8,6],[3,5,8,7],[4,6,9,9],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,19,11,20],[8,1,9,2],[18,7,19,8],[11,3,12,2],[12,17,13,18],[13,6,14,7],[3,14,4,15],[5,16,6,17],[4,16,5,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,3,-11,-4)(15,4,-16,-5)(17,6,-18,-7)(20,11,-1,-12)(12,1,-13,-2)(2,13,-3,-14)(9,14,-10,-15)(7,16,-8,-17)(5,18,-6,-19)(19,8,-20,-9)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12)(-2,-14,9,-20,-12)(-3,10,14)(-4,15,-10)(-5,-19,-9,-15)(-6,17,-8,19)(-7,-17)(-11,20,8,16,4)(-13,2)(-16,7,-18,5)(1,11,3,13)(6,18)
Loop annotated with half-edges
12^1_70 annotated with half-edges