$11^{1}_{100}$ - Minimal pinning sets
Pinning sets for 11^1_100 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_100 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 240
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97978
on average over minimal pinning sets: 2.375
on average over optimal pinning sets: 2.375
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 10}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {2, 4, 5, 10}
4
[2, 2, 2, 4]
2.50
C (optimal)
• {2, 4, 8, 10}
4
[2, 2, 2, 4]
2.50
D (optimal)
• {2, 4, 6, 10}
4
[2, 2, 2, 3]
2.25
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
4
0
0
2.38
5
0
0
22
2.67
6
0
0
52
2.87
7
0
0
69
3.0
8
0
0
56
3.09
9
0
0
28
3.17
10
0
0
8
3.22
11
0
0
1
3.27
Total
4
0
236
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,5,6,0],[0,6,7,1],[1,7,7,5],[1,4,8,2],[2,8,8,3],[3,8,4,4],[5,7,6,6]]
PD code (use to draw this loop with SnapPy): [[5,18,6,1],[13,4,14,5],[17,6,18,7],[1,12,2,13],[3,8,4,9],[14,8,15,7],[11,16,12,17],[2,10,3,9],[15,10,16,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(14,3,-15,-4)(4,11,-5,-12)(18,5,-1,-6)(6,17,-7,-18)(12,7,-13,-8)(16,9,-17,-10)(8,13,-9,-14)(2,15,-3,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-17,6)(-2,-16,-10)(-3,14,-9,16)(-4,-12,-8,-14)(-5,18,-7,12)(-6,-18)(-11,4,-15,2)(-13,8)(1,5,11)(3,15)(7,17,9,13)
Loop annotated with half-edges
11^1_100 annotated with half-edges