$12^{1}_{233}$ - Minimal pinning sets
Pinning sets for 12^1_233 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_233 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 96
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: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 6, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 4, 5, 6, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,7,7,8],[0,8,8,4],[0,3,8,5],[1,4,6,1],[1,5,9,9],[2,9,9,2],[2,4,3,3],[6,7,7,6]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[15,6,16,7],[19,10,20,11],[8,3,9,4],[1,4,2,5],[5,14,6,15],[16,14,17,13],[11,18,12,19],[2,9,3,10],[17,12,18,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(19,2,-20,-3)(13,4,-14,-5)(20,7,-1,-8)(17,8,-18,-9)(15,10,-16,-11)(11,14,-12,-15)(3,12,-4,-13)(9,16,-10,-17)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,18,8)(-2,19,-6)(-3,-13,-5,-19)(-4,13)(-7,20,2)(-8,17,-10,15,-12,3,-20)(-9,-17)(-11,-15)(-14,11,-16,9,-18,5)(1,7)(4,12,14)(10,16)
Loop annotated with half-edges
12^1_233 annotated with half-edges