$11^{3}_{18}$ - Minimal pinning sets
Pinning sets for 11^3_18 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^3_18 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 224
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04496
on average over minimal pinning sets: 2.5
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, 4, 6, 10}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {2, 3, 6, 10}
4
[2, 2, 3, 3]
2.50
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
14
2.74
6
0
0
41
2.91
7
0
0
65
3.03
8
0
0
60
3.12
9
0
0
32
3.19
10
0
0
9
3.24
11
0
0
1
3.27
Total
2
0
222
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,6],[0,6,6,7],[0,7,5,1],[1,4,8,2],[2,8,3,3],[3,8,8,4],[5,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[4,14,1,5],[5,10,6,11],[11,3,12,4],[13,18,14,15],[1,9,2,10],[6,2,7,3],[12,16,13,15],[8,17,9,18],[7,17,8,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(10,1,-11,-2)(3,8,-4,-9)(14,9,-5,-10)(2,13,-3,-14)(15,6,-16,-7)(7,16,-8,-17)(12,17,-13,-18)(18,11,-15,-12)
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)(-17,17)(-18,18)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-5)(-2,-14,-10)(-3,-9,14)(-4,5,9)(-6,15,11,1)(-7,-17,12,-15)(-8,3,13,17)(-11,18,-13,2)(-12,-18)(-16,7)(4,8,16,6)
Multiloop annotated with half-edges
11^3_18 annotated with half-edges