$10^{2}_{19}$ - Minimal pinning sets
Pinning sets for 10^2_19 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^2_19 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 84
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: 2.97951
on average over minimal pinning sets: 2.61333
on average over optimal pinning sets: 2.6
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, 3, 3, 3]
2.60
B (optimal)
• {1, 3, 4, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 2, 4, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
D (optimal)
• {1, 3, 4, 7, 8}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 2, 4, 6, 7, 9}
6
[2, 2, 3, 3, 3, 3]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
4
0
0
2.6
6
0
1
17
2.81
7
0
0
30
2.98
8
0
0
23
3.1
9
0
0
8
3.17
10
0
0
1
3.2
Total
4
1
79
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,5,2],[0,1,5,0],[0,6,7,7],[1,7,6,5],[1,4,6,2],[3,5,4,7],[3,6,4,3]]
PD code (use to draw this multiloop with SnapPy): [[6,16,1,7],[7,14,8,15],[15,5,16,6],[1,12,2,11],[3,13,4,14],[8,4,9,5],[12,9,13,10],[2,10,3,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,2,-16,-3)(12,3,-13,-4)(8,5,-9,-6)(1,10,-2,-11)(14,11,-15,-12)(4,13,-5,-14)(9,16,-10,-7)(6,7,-1,-8)
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,-11,14,-5,8)(-2,15,11)(-3,12,-15)(-4,-14,-12)(-6,-8)(-7,6,-9)(-10,1,7)(-13,4)(-16,9,5,13,3)(2,10,16)
Multiloop annotated with half-edges
10^2_19 annotated with half-edges