$10^{1}_{12}$ - Minimal pinning sets
Pinning sets for 10^1_12 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_12 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 48
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: 2.8307
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, 3, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 5, 9}
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
9
2.56
7
0
0
16
2.8
8
0
0
14
2.98
9
0
0
6
3.11
10
0
0
1
3.2
Total
2
0
46
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,3],[0,2,7,7],[0,7,7,6],[1,6,6,1],[2,5,5,4],[3,4,4,3]]
PD code (use to draw this loop with SnapPy): [[16,11,1,12],[12,6,13,5],[15,4,16,5],[10,3,11,4],[1,9,2,8],[6,14,7,13],[7,14,8,15],[2,9,3,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(13,2,-14,-3)(3,10,-4,-11)(6,15,-7,-16)(16,7,-1,-8)(14,9,-15,-10)(11,4,-12,-5)(5,12,-6,-13)
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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,13,-6,-16,-8)(-3,-11,-5,-13)(-4,11)(-7,16)(-9,14,2)(-10,3,-14)(-12,5)(-15,6,12,4,10)(1,7,15,9)
Loop annotated with half-edges
10^1_12 annotated with half-edges