$10^{3}_{7}$ - Minimal pinning sets
Pinning sets for 10^3_7 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^3_7 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 120
of which optimal: 2
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9873
on average over minimal pinning sets: 2.75
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 7, 9}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 4, 5, 9}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {2, 4, 5, 8, 9, 10}
6
[2, 2, 3, 3, 4, 4]
3.00
b (minimal)
• {1, 3, 4, 6, 7, 9}
6
[2, 2, 3, 3, 4, 4]
3.00
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
2.5
5
0
0
12
2.73
6
0
2
29
2.9
7
0
0
40
3.03
8
0
0
26
3.12
9
0
0
8
3.17
10
0
0
1
3.2
Total
2
2
116
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,3],[0,2,7,7],[0,7,6,5],[1,4,6,1],[2,5,4,7],[3,6,4,3]]
PD code (use to draw this multiloop with SnapPy): [[4,10,1,5],[5,11,6,16],[3,15,4,16],[9,14,10,15],[1,8,2,7],[11,7,12,6],[12,2,13,3],[13,8,14,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(13,10,-14,-5)(14,3,-15,-4)(6,15,-7,-16)(2,11,-3,-12)(16,7,-11,-8)(1,8,-2,-9)(9,12,-10,-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,-9,-13,-5)(-2,-12,9)(-3,14,10,12)(-4,5,-14)(-6,-16,-8,1)(-7,16)(-10,13)(-11,2,8)(-15,6,4)(3,11,7,15)
Multiloop annotated with half-edges
10^3_7 annotated with half-edges