$12^{1}_{207}$ - Minimal pinning sets
Pinning sets for 12^1_207 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_207 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 192
of which optimal: 6
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03439
on average over minimal pinning sets: 2.5
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 7, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
B (optimal)
• {1, 3, 4, 7, 8, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
C (optimal)
• {1, 3, 5, 7, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
D (optimal)
• {1, 3, 5, 7, 8, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
E (optimal)
• {1, 2, 4, 7, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
F (optimal)
• {1, 2, 4, 7, 8, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
6
0
0
2.5
7
0
0
29
2.78
8
0
0
57
2.98
9
0
0
58
3.12
10
0
0
32
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
6
0
186
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 3, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,3],[0,2,6,7],[0,8,9,9],[1,9,8,1],[2,7,7,3],[3,6,6,8],[4,7,5,9],[4,8,5,4]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[4,11,5,12],[19,10,20,11],[6,10,7,9],[1,14,2,15],[12,3,13,4],[18,7,19,8],[8,17,9,18],[13,16,14,17],[2,16,3,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,3,-15,-4)(5,2,-6,-3)(15,6,-16,-7)(20,7,-1,-8)(17,10,-18,-11)(11,18,-12,-19)(9,12,-10,-13)(4,13,-5,-14)(1,16,-2,-17)(8,19,-9,-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,-17,-11,-19,8)(-2,5,13,-10,17)(-3,14,-5)(-4,-14)(-6,15,3)(-7,20,-9,-13,4,-15)(-8,-20)(-12,9,19)(-16,1,7)(-18,11)(2,16,6)(10,12,18)
Loop annotated with half-edges
12^1_207 annotated with half-edges