$12^{1}_{218}$ - Minimal pinning sets
Pinning sets for 12^1_218 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_218 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, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 5, 8, 11}
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],[0,7,7,8],[0,8,8,5],[1,4,2,1],[2,9,9,7],[3,6,9,3],[3,9,4,4],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,18,11,17],[19,16,20,17],[8,3,9,4],[1,13,2,12],[18,12,19,11],[6,15,7,16],[4,7,5,8],[2,13,3,14],[14,5,15,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(12,3,-13,-4)(2,5,-3,-6)(18,7,-19,-8)(9,16,-10,-17)(10,19,-11,-20)(20,11,-1,-12)(4,13,-5,-14)(17,14,-18,-15)(15,8,-16,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,-3,12)(-2,-6)(-4,-14,17,-10,-20,-12)(-5,2,-7,18,14)(-8,15,-18)(-9,-17,-15)(-11,20)(-13,4)(-16,9)(-19,10,16,8)(1,11,19,7)(3,5,13)
Loop annotated with half-edges
12^1_218 annotated with half-edges