$11^{3}_{5}$ - Minimal pinning sets
Pinning sets for 11^3_5 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^3_5 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 60
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.85309
on average over minimal pinning sets: 2.25
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, 7, 8, 10}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 4, 6, 7, 8, 10}
6
[2, 2, 2, 2, 2, 4]
2.33
C (optimal)
• {1, 3, 4, 7, 8, 10}
6
[2, 2, 2, 2, 2, 4]
2.33
D (optimal)
• {1, 4, 5, 7, 8, 10}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
4
0
0
2.25
7
0
0
14
2.63
8
0
0
20
2.88
9
0
0
15
3.04
10
0
0
6
3.17
11
0
0
1
3.27
Total
4
0
56
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,4,0],[0,5,5,6],[0,6,7,1],[1,7,8,8],[2,8,8,2],[2,7,7,3],[3,6,6,4],[4,5,5,4]]
PD code (use to draw this multiloop with SnapPy): [[6,12,1,7],[7,5,8,6],[11,18,12,13],[1,4,2,5],[8,16,9,15],[13,10,14,11],[3,17,4,18],[2,17,3,16],[9,14,10,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,2,-6,-3)(14,3,-15,-4)(10,17,-11,-18)(4,13,-5,-14)(12,15,-7,-16)(7,6,-8,-1)(1,8,-2,-9)(18,9,-13,-10)(16,11,-17,-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,-9,18,-11,16,-7)(-2,5,13,9)(-3,14,-5)(-4,-14)(-6,7,15,3)(-8,1)(-10,-18)(-12,-16)(-13,4,-15,12,-17,10)(2,8,6)(11,17)
Multiloop annotated with half-edges
11^3_5 annotated with half-edges