$10^{3}_{1}$ - Minimal pinning sets
Pinning sets for 10^3_1 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^3_1 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 122
of which optimal: 4
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91136
on average over minimal pinning sets: 2.48
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)
• {1, 3, 8, 10}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 2, 3, 8}
4
[2, 2, 2, 3]
2.25
C (optimal)
• {1, 3, 8, 9}
4
[2, 2, 2, 5]
2.75
D (optimal)
• {1, 3, 6, 8}
4
[2, 2, 2, 5]
2.75
a (minimal)
• {1, 3, 4, 5, 8}
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
4
4
0
0
2.5
5
0
1
18
2.69
6
0
0
35
2.86
7
0
0
35
2.98
8
0
0
21
3.07
9
0
0
7
3.14
10
0
0
1
3.2
Total
4
1
117
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],[2,7,5,5],[2,4,4,6],[3,5,7,7],[3,6,6,4]]
PD code (use to draw this multiloop with SnapPy): [[6,12,1,7],[7,5,8,6],[8,11,9,12],[1,4,2,5],[10,16,11,13],[9,16,10,15],[3,14,4,15],[2,14,3,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,2,-6,-3)(12,3,-13,-4)(13,16,-14,-11)(4,11,-5,-12)(7,6,-8,-1)(1,8,-2,-9)(9,14,-10,-15)(15,10,-16,-7)
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,-15,-7)(-2,5,11,-14,9)(-3,12,-5)(-4,-12)(-6,7,-16,13,3)(-8,1)(-10,15)(-11,4,-13)(2,8,6)(10,14,16)
Multiloop annotated with half-edges
10^3_1 annotated with half-edges