$11^{1}_{72}$ - Minimal pinning sets
Pinning sets for 11^1_72 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_72 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 112
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97993
on average over minimal pinning sets: 2.51333
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, 3, 5, 6, 10}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 3, 4, 5, 8, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 3, 5, 7, 8, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
c (minimal)
• {1, 2, 4, 5, 8, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
d (minimal)
• {1, 2, 4, 5, 6, 10}
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
5
1
0
0
2.4
6
0
4
6
2.62
7
0
0
29
2.85
8
0
0
38
3.02
9
0
0
25
3.15
10
0
0
8
3.23
11
0
0
1
3.27
Total
1
4
107
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,7],[0,7,7,3],[0,2,4,4],[0,3,3,8],[1,8,8,6],[1,5,8,7],[1,6,2,2],[4,6,5,5]]
PD code (use to draw this loop with SnapPy): [[18,3,1,4],[4,15,5,16],[10,17,11,18],[11,2,12,3],[1,12,2,13],[7,14,8,15],[5,8,6,9],[16,9,17,10],[13,6,14,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,18,-10,-1)(16,1,-17,-2)(11,4,-12,-5)(3,6,-4,-7)(14,7,-15,-8)(17,10,-18,-11)(5,12,-6,-13)(2,13,-3,-14)(8,15,-9,-16)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-9)(-2,-14,-8,-16)(-3,-7,14)(-4,11,-18,9,15,7)(-5,-13,2,-17,-11)(-6,3,13)(-10,17,1)(-12,5)(-15,8)(4,6,12)(10,18)
Loop annotated with half-edges
11^1_72 annotated with half-edges