$12^{1}_{62}$ - Minimal pinning sets
Pinning sets for 12^1_62 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_62 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 240
of which optimal: 2
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03455
on average over minimal pinning sets: 2.45
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 10, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 5, 10, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 3, 4, 7, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 3, 5, 7, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
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
2.4
6
0
2
13
2.66
7
0
0
45
2.86
8
0
0
71
3.02
9
0
0
64
3.14
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
2
236
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,2],[0,1,5,0],[0,6,7,1],[1,8,5,5],[2,4,4,6],[3,5,9,7],[3,6,9,8],[4,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,15,1,16],[16,13,17,14],[14,19,15,20],[1,12,2,13],[17,9,18,8],[18,7,19,8],[11,6,12,7],[2,6,3,5],[9,5,10,4],[10,3,11,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,5,-1,-6)(1,18,-2,-19)(7,2,-8,-3)(14,3,-15,-4)(4,13,-5,-14)(17,8,-18,-9)(12,9,-13,-10)(16,11,-17,-12)(10,15,-11,-16)(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,-19,6)(-2,7,19)(-3,14,-5,20,-7)(-4,-14)(-6,-20)(-8,17,11,15,3)(-9,12,-17)(-10,-16,-12)(-11,16)(-13,4,-15,10)(-18,1,5,13,9)(2,18,8)
Loop annotated with half-edges
12^1_62 annotated with half-edges