$10^{4}_{1}$ - Minimal pinning sets
Pinning sets for 10^4_1 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^4_1 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 64
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.8189
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 8}
4
[2, 2, 2, 2]
2.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.0
5
0
0
6
2.4
6
0
0
15
2.67
7
0
0
20
2.86
8
0
0
15
3.0
9
0
0
6
3.11
10
0
0
1
3.2
Total
1
0
63
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 4, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,5,6,6],[0,6,4,0],[1,3,7,1],[1,7,7,2],[2,7,3,2],[4,6,5,5]]
PD code (use to draw this multiloop with SnapPy): [[4,8,1,5],[5,9,6,12],[3,16,4,13],[7,1,8,2],[9,7,10,6],[11,13,12,14],[15,2,16,3],[10,15,11,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(16,3,-13,-4)(15,12,-16,-9)(2,13,-3,-14)(11,14,-12,-15)(8,9,-5,-10)(1,6,-2,-7)(10,7,-11,-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,-7,10,-5)(-2,-14,11,7)(-3,16,12,14)(-4,5,9,-16)(-6,1)(-8,-10)(-9,8,-11,-15)(-12,15)(-13,2,6,4)(3,13)
Multiloop annotated with half-edges
10^4_1 annotated with half-edges