$10^{1}_{16}$ - Minimal pinning sets
Pinning sets for 10^1_16 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_16 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 64
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90413
on average over minimal pinning sets: 2.4
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, 4, 6, 7}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 5, 6, 7}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 4, 6, 7}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.4
6
0
0
13
2.69
7
0
0
22
2.9
8
0
0
18
3.04
9
0
0
7
3.14
10
0
0
1
3.2
Total
3
0
61
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,2],[0,1,5,6],[0,6,6,0],[1,7,7,5],[1,4,7,2],[2,7,3,3],[4,6,5,4]]
PD code (use to draw this loop with SnapPy): [[16,9,1,10],[10,5,11,6],[6,15,7,16],[8,1,9,2],[13,4,14,5],[11,14,12,15],[7,3,8,2],[3,12,4,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(7,2,-8,-3)(13,6,-14,-7)(16,9,-1,-10)(8,11,-9,-12)(3,12,-4,-13)(5,14,-6,-15)(15,4,-16,-5)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,7,-14,5,-16,-10)(-3,-13,-7)(-4,15,-6,13)(-5,-15)(-8,-12,3)(-9,16,4,12)(-11,8,2)(1,9,11)(6,14)
Loop annotated with half-edges
10^1_16 annotated with half-edges