$11^{2}_{9}$ - Minimal pinning sets
Pinning sets for 11^2_9 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_9 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 160
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97092
on average over minimal pinning sets: 2.325
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 6}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 4, 6, 10}
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
1
0
0
2.25
5
0
1
7
2.55
6
0
0
26
2.77
7
0
0
45
2.93
8
0
0
45
3.06
9
0
0
26
3.15
10
0
0
8
3.23
11
0
0
1
3.27
Total
1
1
158
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,7,7,8],[0,8,8,6],[0,5,5,1],[1,4,4,6],[1,5,3,7],[2,6,8,2],[2,7,3,3]]
PD code (use to draw this multiloop with SnapPy): [[6,18,1,7],[7,10,8,11],[15,5,16,6],[17,12,18,13],[1,9,2,10],[8,2,9,3],[11,3,12,4],[4,14,5,15],[16,14,17,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(10,1,-11,-2)(14,3,-15,-4)(5,8,-6,-9)(18,9,-7,-10)(16,11,-17,-12)(12,15,-13,-16)(2,13,-3,-14)(4,17,-5,-18)
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,-7)(-2,-14,-4,-18,-10)(-3,14)(-5,-9,18)(-6,7,9)(-8,5,17,11,1)(-11,16,-13,2)(-12,-16)(-15,12,-17,4)(3,13,15)(6,8)
Multiloop annotated with half-edges
11^2_9 annotated with half-edges