$12^{3}_{20}$ - Minimal pinning sets
Pinning sets for 12^3_20
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_20
Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 382
of which optimal: 1
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03785
on average over minimal pinning sets: 2.55
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, 7, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
•
{1, 2, 7, 8, 10}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
•
{1, 2, 6, 7, 10}
5
[2, 2, 2, 3, 4]
2.60
c (minimal)
•
{1, 2, 7, 10, 12}
5
[2, 2, 2, 3, 4]
2.60
d (minimal)
•
{1, 2, 5, 7, 10}
5
[2, 2, 2, 3, 4]
2.60
e (minimal)
•
{1, 2, 7, 10, 11}
5
[2, 2, 2, 3, 5]
2.80
f (minimal)
•
{1, 2, 4, 7, 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
2.25
5
0
6
8
2.59
6
0
0
49
2.81
7
0
0
91
2.97
8
0
0
105
3.08
9
0
0
77
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
6
375
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,6],[0,6,4,4],[1,3,3,7],[2,7,8,9],[2,9,7,3],[4,6,8,5],[5,7,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[4,14,1,5],[5,3,6,4],[6,13,7,14],[1,15,2,20],[2,19,3,20],[9,12,10,13],[7,16,8,15],[8,18,9,19],[11,17,12,18],[10,17,11,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(16,19,-17,-20)(13,20,-14,-11)(2,11,-3,-12)(8,17,-9,-18)(18,9,-19,-10)(15,10,-16,-5)(4,5,-1,-6)(6,3,-7,-4)(7,14,-8,-15)
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,-3,6)(-2,-12)(-4,-6)(-5,4,-7,-15)(-8,-18,-10,15)(-9,18)(-11,2,-13)(-14,7,3,11)(-16,-20,13,1,5)(-17,8,14,20)(-19,16,10)(9,17,19)
Multiloop annotated with half-edges
12^3_20 annotated with half-edges