$12^{2}_{90}$ - Minimal pinning sets
Pinning sets for 12^2_90
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_90
Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 432
of which optimal: 1
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05593
on average over minimal pinning sets: 2.5125
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)
•
{3, 6, 8, 9}
4
[2, 2, 2, 4]
2.50
a (minimal)
•
{1, 3, 6, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
•
{3, 4, 6, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
c (minimal)
•
{3, 6, 7, 9, 11}
5
[2, 2, 2, 3, 5]
2.80
d (minimal)
•
{1, 3, 6, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
e (minimal)
•
{3, 6, 7, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
f (minimal)
•
{3, 5, 6, 9, 10}
5
[2, 2, 2, 3, 5]
2.80
g (minimal)
•
{3, 4, 6, 9, 10}
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
1
0
0
2.5
5
0
7
8
2.64
6
0
0
58
2.84
7
0
0
109
3.0
8
0
0
120
3.1
9
0
0
83
3.18
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
7
424
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,7,3],[0,2,8,9],[0,5,1,1],[1,4,9,6],[2,5,7,7],[2,6,6,8],[3,7,9,9],[3,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[3,12,4,1],[2,20,3,13],[11,8,12,9],[4,8,5,7],[1,14,2,13],[14,19,15,20],[9,15,10,16],[16,10,17,11],[5,17,6,18],[18,6,19,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (19,4,-20,-5)(5,2,-6,-3)(9,6,-10,-7)(17,8,-18,-9)(1,10,-2,-11)(7,16,-8,-17)(3,18,-4,-19)(15,20,-16,-13)(12,13,-1,-14)(14,11,-15,-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)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-11,14)(-2,5,-20,15,11)(-3,-19,-5)(-4,19)(-6,9,-18,3)(-7,-17,-9)(-8,17)(-10,1,13,-16,7)(-12,-14)(-13,12,-15)(2,10,6)(4,18,8,16,20)
Multiloop annotated with half-edges
12^2_90 annotated with half-edges