$11^{2}_{67}$ - Minimal pinning sets
Pinning sets for 11^2_67 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_67 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 324
of which optimal: 4
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.13738
on average over minimal pinning sets: 2.9
on average over optimal pinning sets: 3.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 8, 11}
4
[2, 3, 3, 3]
2.75
B (optimal)
• {2, 4, 9, 11}
4
[2, 3, 3, 5]
3.25
C (optimal)
• {1, 2, 4, 11}
4
[2, 3, 3, 3]
2.75
D (optimal)
• {2, 4, 6, 11}
4
[2, 3, 3, 5]
3.25
a (minimal)
• {2, 4, 5, 7, 11}
5
[2, 3, 3, 3, 3]
2.80
b (minimal)
• {1, 3, 4, 8, 10}
5
[2, 3, 3, 3, 3]
2.80
c (minimal)
• {1, 3, 4, 10, 11}
5
[2, 3, 3, 3, 3]
2.80
d (minimal)
• {2, 3, 4, 8, 10}
5
[2, 3, 3, 3, 3]
2.80
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
3.0
5
0
4
22
3.02
6
0
0
68
3.07
7
0
0
96
3.13
8
0
0
80
3.19
9
0
0
39
3.23
10
0
0
10
3.26
11
0
0
1
3.27
Total
4
4
316
Other information about this multiloop Properties
Region degree sequence: [2, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,3],[0,2,6,7],[0,8,5,1],[1,4,8,2],[3,8,7,7],[3,6,6,8],[4,7,6,5]]
PD code (use to draw this multiloop with SnapPy): [[4,18,1,5],[5,12,6,13],[13,3,14,4],[14,17,15,18],[1,11,2,12],[6,2,7,3],[16,9,17,10],[15,9,16,8],[10,7,11,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(12,1,-13,-2)(7,14,-8,-15)(15,8,-16,-9)(3,10,-4,-11)(18,11,-5,-12)(13,16,-14,-17)(2,17,-3,-18)(9,6,-10,-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)(-17,17)(-18,18)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-5)(-2,-18,-12)(-3,-11,18)(-4,5,11)(-6,9,-16,13,1)(-7,-15,-9)(-8,15)(-10,3,17,-14,7)(-13,-17,2)(4,10,6)(8,14,16)
Multiloop annotated with half-edges
11^2_67 annotated with half-edges