$12^{4}_{9}$ - Minimal pinning sets
Pinning sets for 12^4_9
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^4_9
Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 124
of which optimal: 5
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91549
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{1, 2, 5, 7, 9, 12}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
•
{1, 2, 5, 7, 10, 12}
6
[2, 2, 2, 2, 2, 4]
2.33
C (optimal)
•
{1, 2, 5, 6, 7, 12}
6
[2, 2, 2, 2, 2, 4]
2.33
D (optimal)
•
{1, 2, 4, 5, 7, 12}
6
[2, 2, 2, 2, 2, 4]
2.33
E (optimal)
•
{1, 2, 3, 5, 7, 12}
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
5
0
0
2.27
7
0
0
20
2.63
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
5
0
119
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,5,6],[0,6,7,7],[1,8,8,5],[1,4,2,2],[2,9,9,3],[3,9,8,3],[4,7,9,4],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[6,10,1,7],[7,5,8,6],[9,16,10,11],[1,17,2,20],[4,12,5,13],[8,12,9,11],[15,17,16,18],[2,19,3,20],[13,3,14,4],[18,14,19,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(18,3,-19,-4)(4,17,-5,-18)(2,11,-3,-12)(10,13,-7,-14)(20,15,-17,-16)(16,19,-11,-20)(6,7,-1,-8)(8,5,-9,-6)(14,9,-15,-10)
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)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-3,18,-5,8)(-2,-12)(-4,-18)(-6,-8)(-7,6,-9,14)(-10,-14)(-11,2,-13,10,-15,20)(-16,-20)(-17,4,-19,16)(1,7,13)(3,11,19)(5,17,15,9)
Multiloop annotated with half-edges
12^4_9 annotated with half-edges