$10^{2}_{7}$ - Minimal pinning sets
Pinning sets for 10^2_7 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^2_7 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 109
of which optimal: 2
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91507
on average over minimal pinning sets: 2.63333
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 6, 7}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 4, 7, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 4, 7, 8, 10}
5
[2, 2, 2, 4, 5]
3.00
b (minimal)
• {1, 4, 5, 7, 10}
5
[2, 2, 2, 4, 5]
3.00
c (minimal)
• {1, 2, 3, 4, 7, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.25
5
0
2
11
2.65
6
0
1
30
2.84
7
0
0
34
2.97
8
0
0
21
3.07
9
0
0
7
3.14
10
0
0
1
3.2
Total
2
3
104
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,5],[0,6,7,1],[2,7,7,5],[2,4,6,6],[3,5,5,7],[3,6,4,4]]
PD code (use to draw this multiloop with SnapPy): [[12,5,1,6],[6,11,7,12],[7,4,8,5],[1,10,2,11],[3,16,4,13],[8,16,9,15],[9,14,10,15],[2,14,3,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,2,-6,-3)(10,3,-11,-4)(11,8,-12,-9)(4,9,-5,-10)(15,6,-16,-7)(1,16,-2,-13)(13,12,-14,-1)(7,14,-8,-15)
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)(-2,5,9,-12,13)(-3,10,-5)(-4,-10)(-6,15,-8,11,3)(-7,-15)(-9,4,-11)(-14,7,-16,1)(2,16,6)(8,14,12)
Multiloop annotated with half-edges
10^2_7 annotated with half-edges