$12^{1}_{344}$ - Minimal pinning sets
Pinning sets for 12^1_344 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_344 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 280
of which optimal: 3
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04711
on average over minimal pinning sets: 2.51667
on average over optimal pinning sets: 2.46667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 5, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 5, 10, 11}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 2, 4, 5, 8, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.47
6
0
1
20
2.71
7
0
0
58
2.91
8
0
0
84
3.05
9
0
0
70
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
1
276
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: [[1,2,3,4],[0,4,5,2],[0,1,5,6],[0,7,7,8],[0,8,9,1],[1,9,9,2],[2,8,7,7],[3,6,6,3],[3,6,9,4],[4,8,5,5]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[11,4,12,5],[12,19,13,20],[6,15,7,16],[1,10,2,11],[18,3,19,4],[13,9,14,8],[14,7,15,8],[16,9,17,10],[2,17,3,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,20,-16,-1)(8,1,-9,-2)(11,4,-12,-5)(5,10,-6,-11)(13,6,-14,-7)(18,7,-19,-8)(3,12,-4,-13)(19,14,-20,-15)(9,16,-10,-17)(2,17,-3,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,-19,-15)(-2,-18,-8)(-3,-13,-7,18)(-4,11,-6,13)(-5,-11)(-9,-17,2)(-10,5,-12,3,17)(-14,19,7)(-16,9,1)(-20,15)(4,12)(6,10,16,20,14)
Loop annotated with half-edges
12^1_344 annotated with half-edges