$12^{1}_{267}$ - Minimal pinning sets
Pinning sets for 12^1_267 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_267 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 140
of which optimal: 3
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97615
on average over minimal pinning sets: 2.42857
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 6, 7, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 3, 4, 6, 7, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 3, 4, 6, 7, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
a (minimal)
• {1, 2, 4, 6, 7, 10, 11}
7
[2, 2, 2, 2, 3, 3, 4]
2.57
b (minimal)
• {1, 2, 4, 5, 6, 7, 10}
7
[2, 2, 2, 2, 3, 3, 4]
2.57
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
3
0
0
2.33
7
0
2
16
2.63
8
0
0
40
2.88
9
0
0
44
3.06
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
2
135
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,5],[0,6,7,3],[0,2,7,0],[1,8,9,5],[1,4,9,1],[2,9,9,8],[2,8,8,3],[4,7,7,6],[4,6,6,5]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[6,13,7,14],[19,2,20,3],[8,2,9,1],[14,11,15,12],[12,5,13,6],[3,16,4,17],[18,9,19,10],[10,17,11,18],[15,4,16,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(9,2,-10,-3)(15,6,-16,-7)(11,8,-12,-9)(3,10,-4,-11)(20,13,-1,-14)(7,14,-8,-15)(4,17,-5,-18)(18,5,-19,-6)(16,19,-17,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,8,14)(-2,9,-12)(-3,-11,-9)(-4,-18,-6,15,-8,11)(-5,18)(-7,-15)(-10,3)(-13,20,-17,4,10,2)(-14,7,-16,-20)(-19,16,6)(1,13)(5,17,19)
Loop annotated with half-edges
12^1_267 annotated with half-edges