$12^{1}_{268}$ - Minimal pinning sets
Pinning sets for 12^1_268 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_268 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 152
of which optimal: 5
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97907
on average over minimal pinning sets: 2.4
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 6, 7, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 3, 5, 6, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 2, 3, 7, 8, 9}
6
[2, 2, 2, 2, 3, 4]
2.50
D (optimal)
• {1, 2, 3, 7, 9, 10}
6
[2, 2, 2, 2, 3, 4]
2.50
E (optimal)
• {1, 2, 3, 7, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
5
0
0
2.4
7
0
0
23
2.7
8
0
0
44
2.91
9
0
0
45
3.07
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
5
0
147
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: [[1,2,3,4],[0,5,5,2],[0,1,5,3],[0,2,6,6],[0,6,7,7],[1,8,2,1],[3,9,4,3],[4,9,8,4],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,18,11,17],[19,16,20,17],[8,15,9,16],[1,7,2,6],[18,12,19,11],[14,7,15,8],[2,5,3,6],[12,3,13,4],[4,13,5,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(19,2,-20,-3)(3,18,-4,-19)(11,4,-12,-5)(17,6,-18,-7)(8,15,-9,-16)(1,10,-2,-11)(5,12,-6,-13)(16,13,-17,-14)(14,7,-15,-8)
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,-11,-5,-13,16,-9)(-2,19,-4,11)(-3,-19)(-6,17,13)(-7,14,-17)(-8,-16,-14)(-10,1)(-12,5)(-15,8)(-18,3,-20,9,15,7)(2,10,20)(4,18,6,12)
Loop annotated with half-edges
12^1_268 annotated with half-edges