$10^{2}_{22}$ - Minimal pinning sets
Pinning sets for 10^2_22 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^2_22 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 56
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: 2.83929
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 6, 7, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 7, 9}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 3, 7, 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
3
0
0
2.27
6
0
0
12
2.61
7
0
0
19
2.84
8
0
0
15
3.0
9
0
0
6
3.11
10
0
0
1
3.2
Total
3
0
53
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,4,5,3],[0,2,6,6],[1,7,2,1],[2,7,7,6],[3,5,7,3],[4,6,5,5]]
PD code (use to draw this multiloop with SnapPy): [[10,16,1,11],[11,9,12,10],[15,7,16,8],[1,7,2,6],[8,12,9,13],[4,14,5,15],[2,5,3,6],[13,3,14,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,10,-12,-1)(9,2,-10,-3)(3,8,-4,-9)(13,4,-14,-5)(15,6,-16,-7)(1,12,-2,-13)(7,14,-8,-15)(5,16,-6,-11)
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,-13,-5,-11)(-2,9,-4,13)(-3,-9)(-6,15,-8,3,-10,11)(-7,-15)(-12,1)(-14,7,-16,5)(2,12,10)(4,8,14)(6,16)
Multiloop annotated with half-edges
10^2_22 annotated with half-edges