- Minimal pinning sets
Pinning sets for 12^3_16 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_16 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
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: 2.96906
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 6, 9, 10}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 6, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,6,7,8],[0,9,4,4],[1,3,3,9],[1,9,6,6],[2,5,5,7],[2,6,8,8],[2,7,7,9],[3,8,5,4]]
PD code (use to draw this multiloop with SnapPy): [[4,12,1,5],[5,3,6,4],[8,11,9,12],[1,13,2,20],[2,19,3,20],[6,14,7,15],[15,7,16,8],[16,10,17,11],[9,17,10,18],[13,18,14,19]]
Permutation representation (action on half-edges):
Vertex permutation (6,3,-7,-4)(17,8,-18,-9)(18,11,-19,-12)(9,12,-10,-5)(4,5,-1,-6)(10,19,-11,-20)(15,20,-16,-13)(2,13,-3,-14)(14,1,-15,-2)(7,16,-8,-17)
Edge permutation (-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)(-19,19)(-20,20)
Face permutation (-1,14,-3,6)(-2,-14)(-4,-6)(-5,4,-7,-17,-9)(-8,17)(-10,-20,15,1,5)(-11,18,8,16,20)(-12,9,-18)(-13,2,-15)(-16,7,3,13)(-19,10,12)(11,19)
Multiloop annotated with half-edges
12^3_16 annotated with half-edges