$12^{1}_{356}$ - Minimal pinning sets
Pinning sets for 12^1_356 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_356 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 224
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9785
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 5, 6, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 5, 6, 9}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 5, 6, 9}
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
3
0
0
2.27
6
0
0
18
2.59
7
0
0
46
2.82
8
0
0
65
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
0
221
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,6,0],[0,7,7,1],[1,8,8,5],[1,4,9,6],[2,5,9,2],[3,9,8,3],[4,7,9,4],[5,8,7,6]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[10,5,11,6],[19,12,20,13],[1,7,2,6],[16,9,17,10],[17,4,18,5],[13,18,14,19],[7,3,8,2],[8,15,9,16],[3,14,4,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(7,2,-8,-3)(3,18,-4,-19)(13,6,-14,-7)(16,9,-17,-10)(20,11,-1,-12)(5,12,-6,-13)(10,15,-11,-16)(8,17,-9,-18)(19,4,-20,-5)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,6,12)(-2,7,-14)(-3,-19,-5,-13,-7)(-4,19)(-6,13)(-8,-18,3)(-9,16,-11,20,4,18)(-10,-16)(-12,5,-20)(-15,10,-17,8,2)(1,11,15)(9,17)
Loop annotated with half-edges
12^1_356 annotated with half-edges