$12^{1}_{123}$ - Minimal pinning sets
Pinning sets for 12^1_123 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_123 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 156
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: 2.97709
on average over minimal pinning sets: 2.41667
on average over optimal pinning sets: 2.41667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 6, 8}
6
[2, 2, 2, 2, 3, 4]
2.50
B (optimal)
• {1, 2, 4, 5, 6, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 2, 3, 5, 6, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
D (optimal)
• {1, 2, 3, 4, 5, 6}
6
[2, 2, 2, 2, 3, 3]
2.33
E (optimal)
• {1, 2, 4, 5, 6, 9}
6
[2, 2, 2, 2, 3, 5]
2.67
F (optimal)
• {1, 2, 4, 5, 6, 7}
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
6
0
0
2.42
7
0
0
25
2.72
8
0
0
45
2.92
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
6
0
150
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 8]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,4],[0,5,5,0],[0,5,6,1],[1,7,8,1],[2,6,3,2],[3,5,9,7],[4,6,9,8],[4,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[11,6,12,7],[19,8,20,9],[1,10,2,11],[5,12,6,13],[9,18,10,19],[2,18,3,17],[13,17,14,16],[4,15,5,16],[3,15,4,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,3,-9,-4)(16,5,-17,-6)(18,7,-19,-8)(2,9,-3,-10)(11,20,-12,-1)(1,12,-2,-13)(13,10,-14,-11)(14,19,-15,-20)(6,15,-7,-16)(4,17,-5,-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,-13,-11)(-2,-10,13)(-3,8,-19,14,10)(-4,-18,-8)(-5,16,-7,18)(-6,-16)(-9,2,12,20,-15,6,-17,4)(-12,1)(-14,-20,11)(3,9)(5,17)(7,15,19)
Loop annotated with half-edges
12^1_123 annotated with half-edges