$11^{2}_{13}$ - Minimal pinning sets
Pinning sets for 11^2_13 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_13 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 250
of which optimal: 5
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97956
on average over minimal pinning sets: 2.48333
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, 6, 8, 10}
4
[2, 2, 2, 5]
2.75
B (optimal)
• {1, 5, 6, 8}
4
[2, 2, 2, 4]
2.50
C (optimal)
• {1, 3, 6, 8}
4
[2, 2, 2, 4]
2.50
D (optimal)
• {1, 6, 7, 8}
4
[2, 2, 2, 3]
2.25
E (optimal)
• {1, 6, 8, 9}
4
[2, 2, 2, 4]
2.50
a (minimal)
• {1, 2, 4, 6, 8}
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
5
0
0
2.5
5
0
1
25
2.71
6
0
0
56
2.88
7
0
0
70
3.0
8
0
0
56
3.09
9
0
0
28
3.17
10
0
0
8
3.23
11
0
0
1
3.27
Total
5
1
244
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,5],[0,5,6,1],[2,6,7,7],[2,7,8,3],[3,8,8,4],[4,8,5,4],[5,7,6,6]]
PD code (use to draw this multiloop with SnapPy): [[7,14,8,1],[13,6,14,7],[8,6,9,5],[1,12,2,13],[9,15,10,18],[11,4,12,5],[2,16,3,15],[10,17,11,18],[3,16,4,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(2,5,-3,-6)(10,3,-11,-4)(12,9,-13,-10)(4,11,-5,-12)(7,16,-8,-17)(17,6,-18,-7)(18,13,-15,-14)(14,15,-1,-16)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,16)(-2,-6,17,-8)(-3,10,-13,18,6)(-4,-12,-10)(-5,2,-9,12)(-7,-17)(-11,4)(-14,-16,7,-18)(-15,14)(1,15,13,9)(3,5,11)
Multiloop annotated with half-edges
11^2_13 annotated with half-edges