$11^{1}_{67}$ - Minimal pinning sets
Pinning sets for 11^1_67 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_67 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90403
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 6, 10}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 7, 10}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
11
2.55
7
0
0
25
2.79
8
0
0
30
2.97
9
0
0
20
3.1
10
0
0
7
3.2
11
0
0
1
3.27
Total
2
0
94
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,3,3],[0,2,2,7],[0,7,5,5],[1,4,4,6],[1,5,8,8],[3,8,8,4],[6,7,7,6]]
PD code (use to draw this loop with SnapPy): [[15,18,16,1],[3,14,4,15],[4,17,5,18],[16,5,17,6],[1,11,2,10],[2,9,3,10],[13,8,14,9],[6,12,7,11],[7,12,8,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,3,-15,-4)(11,6,-12,-7)(18,7,-1,-8)(8,17,-9,-18)(9,4,-10,-5)(5,10,-6,-11)(1,12,-2,-13)(2,15,-3,-16)(13,16,-14,-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,-13,-17,8)(-2,-16,13)(-3,14,16)(-4,9,17,-14)(-5,-11,-7,18,-9)(-6,11)(-8,-18)(-10,5)(-12,1,7)(-15,2,12,6,10,4)(3,15)
Loop annotated with half-edges
11^1_67 annotated with half-edges