$12^{1}_{194}$ - Minimal pinning sets
Pinning sets for 12^1_194 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_194 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 216
of which optimal: 8
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04483
on average over minimal pinning sets: 2.5625
on average over optimal pinning sets: 2.5625
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 6, 8, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
B (optimal)
• {1, 2, 4, 6, 8, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
C (optimal)
• {1, 2, 3, 6, 8, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
D (optimal)
• {1, 2, 3, 6, 9, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
E (optimal)
• {1, 2, 3, 6, 10, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
F (optimal)
• {1, 2, 3, 4, 6, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
G (optimal)
• {1, 2, 4, 5, 6, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
H (optimal)
• {1, 2, 4, 6, 7, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
8
0
0
2.56
7
0
0
36
2.82
8
0
0
66
3.01
9
0
0
63
3.14
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
8
0
208
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,5,5],[0,6,3,0],[0,2,6,4],[1,3,6,7],[1,7,8,1],[2,9,4,3],[4,9,8,5],[5,7,9,9],[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],[1,19,2,18],[10,17,11,18],[5,12,6,13],[9,3,10,2],[16,13,17,14],[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)(2,9,-3,-10)(13,10,-14,-11)(11,20,-12,-1)(1,12,-2,-13)(7,14,-8,-15)(15,18,-16,-19)(4,17,-5,-18)(19,6,-20,-7)
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)(-3,8,14,10)(-4,-18,15,-8)(-5,16,18)(-6,19,-16)(-7,-15,-19)(-9,2,12,20,6,-17,4)(-12,1)(-14,7,-20,11)(3,9)(5,17)
Loop annotated with half-edges
12^1_194 annotated with half-edges