$12^{1}_{188}$ - Minimal pinning sets
Pinning sets for 12^1_188 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_188 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, 7, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 5, 7, 8}
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, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,5,6],[0,7,8,0],[0,5,4,4],[1,3,3,5],[1,4,3,8],[1,9,7,7],[2,6,6,9],[2,9,9,5],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,3,9,4],[6,19,7,20],[1,10,2,11],[11,2,12,3],[9,12,10,13],[4,15,5,16],[16,5,17,6],[18,13,19,14],[14,17,15,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(16,3,-17,-4)(17,6,-18,-7)(5,8,-6,-9)(14,9,-15,-10)(20,11,-1,-12)(10,13,-11,-14)(2,15,-3,-16)(7,18,-8,-19)(4,19,-5,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12)(-2,-16,-4,-20,-12)(-3,16)(-5,-9,14,-11,20)(-6,17,3,15,9)(-7,-19,4,-17)(-8,5,19)(-10,-14)(-13,10,-15,2)(-18,7)(1,11,13)(6,8,18)
Loop annotated with half-edges
12^1_188 annotated with half-edges