$11^{1}_{22}$ - Minimal pinning sets
Pinning sets for 11^1_22 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_22 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 320
of which optimal: 2
of which minimal: 10
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.08734
on average over minimal pinning sets: 2.75
on average over optimal pinning sets: 2.75
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 6, 10}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 5, 7, 10}
4
[2, 2, 4, 4]
3.00
a (minimal)
• {1, 3, 4, 10, 11}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 4, 5, 10, 11}
5
[2, 2, 3, 3, 4]
2.80
c (minimal)
• {1, 2, 6, 7, 10}
5
[2, 2, 3, 3, 4]
2.80
d (minimal)
• {1, 2, 3, 7, 10}
5
[2, 2, 3, 3, 4]
2.80
e (minimal)
• {1, 3, 4, 7, 10}
5
[2, 2, 3, 3, 4]
2.80
f (minimal)
• {1, 2, 6, 8, 10}
5
[2, 2, 3, 3, 3]
2.60
g (minimal)
• {1, 5, 6, 8, 10}
5
[2, 2, 3, 3, 4]
2.80
h (minimal)
• {1, 5, 8, 10, 11}
5
[2, 2, 3, 3, 4]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.75
5
0
8
14
2.85
6
0
0
69
2.99
7
0
0
101
3.09
8
0
0
80
3.16
9
0
0
36
3.21
10
0
0
9
3.24
11
0
0
1
3.27
Total
2
8
310
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,3],[0,2,7,8],[0,5,1,1],[1,4,6,2],[2,5,8,7],[3,6,8,8],[3,7,7,6]]
PD code (use to draw this loop with SnapPy): [[3,18,4,1],[11,2,12,3],[17,8,18,9],[4,8,5,7],[1,10,2,11],[12,10,13,9],[13,16,14,17],[5,14,6,15],[15,6,16,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,18,-10,-1)(13,4,-14,-5)(5,2,-6,-3)(15,6,-16,-7)(7,10,-8,-11)(17,8,-18,-9)(3,12,-4,-13)(11,14,-12,-15)(1,16,-2,-17)
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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-17,-9)(-2,5,-14,11,-8,17)(-3,-13,-5)(-4,13)(-6,15,-12,3)(-7,-11,-15)(-10,7,-16,1)(-18,9)(2,16,6)(4,12,14)(8,10,18)
Loop annotated with half-edges
11^1_22 annotated with half-edges