$12^{1}_{334}$ - Minimal pinning sets
Pinning sets for 12^1_334 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_334 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 320
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03463
on average over minimal pinning sets: 2.325
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 7, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 4, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
1
8
2.56
6
0
0
34
2.77
7
0
0
71
2.94
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
1
318
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,6],[0,6,6,7],[0,8,4,0],[1,3,9,5],[1,4,9,6],[1,5,2,2],[2,9,8,8],[3,7,7,9],[4,8,7,5]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,17,11,18],[19,4,20,5],[8,1,9,2],[16,7,17,8],[11,7,12,6],[18,6,19,5],[3,14,4,15],[2,14,3,13],[15,12,16,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-17,-2)(2,15,-3,-16)(3,8,-4,-9)(13,4,-14,-5)(17,6,-18,-7)(12,9,-13,-10)(10,19,-11,-20)(20,11,-1,-12)(7,14,-8,-15)(5,18,-6,-19)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-3,-9,12)(-2,-16)(-4,13,9)(-5,-19,10,-13)(-6,17,1,11,19)(-7,-15,2,-17)(-8,3,15)(-10,-20,-12)(-11,20)(-14,7,-18,5)(4,8,14)(6,18)
Loop annotated with half-edges
12^1_334 annotated with half-edges