$12^{3}_{21}$ - Minimal pinning sets
Pinning sets for 12^3_21
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_21
Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 435
of which optimal: 1
of which minimal: 10
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05465
on average over minimal pinning sets: 2.59
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)
•
{2, 4, 6, 9}
4
[2, 2, 2, 4]
2.50
a (minimal)
•
{1, 2, 3, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
•
{2, 3, 6, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
c (minimal)
•
{1, 2, 5, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
d (minimal)
•
{2, 5, 6, 9, 12}
5
[2, 2, 2, 3, 4]
2.60
e (minimal)
•
{2, 5, 6, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
f (minimal)
•
{2, 3, 6, 8, 9}
5
[2, 2, 2, 3, 4]
2.60
g (minimal)
•
{2, 5, 6, 8, 9}
5
[2, 2, 2, 3, 4]
2.60
h (minimal)
•
{2, 3, 6, 9, 10, 12}
6
[2, 2, 2, 3, 4, 6]
3.17
i (minimal)
•
{2, 3, 6, 7, 9, 12}
6
[2, 2, 2, 3, 4, 4]
2.83
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
2.63
6
0
2
58
2.84
7
0
0
110
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
9
425
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,3],[0,2,7,8],[0,5,1,1],[1,4,9,2],[2,9,7,7],[3,6,6,8],[3,7,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[3,8,4,1],[2,16,3,9],[11,7,12,8],[4,12,5,13],[1,10,2,9],[10,15,11,16],[6,20,7,17],[5,20,6,19],[13,19,14,18],[14,17,15,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,5,-13,-6)(1,6,-2,-7)(2,13,-3,-14)(11,16,-12,-9)(8,9,-1,-10)(10,7,-11,-8)(19,14,-20,-15)(20,3,-17,-4)(4,17,-5,-18)(15,18,-16,-19)
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,-7,10)(-2,-14,19,-16,11,7)(-3,20,14)(-4,-18,15,-20)(-5,12,16,18)(-6,1,9,-12)(-8,-10)(-9,8,-11)(-13,2,6)(-15,-19)(-17,4)(3,13,5,17)
Multiloop annotated with half-edges
12^3_21 annotated with half-edges