$12^{1}_{222}$ - Minimal pinning sets
Pinning sets for 12^1_222 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_222 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, 3, 5, 8}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 5, 6}
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: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,7,7,4],[0,3,8,5],[1,4,2,1],[2,9,9,7],[3,6,8,3],[4,7,9,9],[6,8,8,6]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,18,9,17],[19,16,20,17],[6,11,7,12],[1,11,2,10],[18,10,19,9],[4,15,5,16],[12,5,13,6],[2,13,3,14],[14,3,15,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(10,3,-11,-4)(18,5,-19,-6)(7,16,-8,-17)(8,19,-9,-20)(4,9,-5,-10)(20,11,-1,-12)(2,13,-3,-14)(17,14,-18,-15)(15,6,-16,-7)
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,-14,17,-8,-20,-12)(-3,10,-5,18,14)(-4,-10)(-6,15,-18)(-7,-17,-15)(-9,4,-11,20)(-13,2)(-16,7)(-19,8,16,6)(1,11,3,13)(5,9,19)
Loop annotated with half-edges
12^1_222 annotated with half-edges