$12^{3}_{44}$ - Minimal pinning sets
Pinning sets for 12^3_44
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_44
Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 248
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: 2.97498
on average over minimal pinning sets: 2.32
on average over optimal pinning sets: 2.32
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{1, 2, 4, 5, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
•
{1, 2, 3, 4, 9}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
•
{1, 2, 4, 8, 9}
5
[2, 2, 2, 2, 4]
2.40
D (optimal)
•
{1, 2, 4, 9, 12}
5
[2, 2, 2, 2, 4]
2.40
E (optimal)
•
{1, 2, 4, 9, 11}
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
5
0
0
2.32
6
0
0
25
2.64
7
0
0
55
2.85
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
5
0
243
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,5],[0,5,6,4],[1,3,7,8],[2,9,3,2],[3,9,7,7],[4,6,6,8],[4,7,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[6,16,1,7],[7,5,8,6],[8,15,9,16],[1,14,2,13],[4,12,5,13],[14,9,15,10],[2,17,3,20],[3,19,4,20],[11,18,12,19],[10,18,11,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,2,-12,-3)(18,13,-19,-14)(1,16,-2,-17)(17,4,-18,-5)(12,19,-13,-20)(3,20,-4,-11)(15,10,-16,-7)(6,7,-1,-8)(8,5,-9,-6)(9,14,-10,-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,-17,-5,8)(-2,11,-4,17)(-3,-11)(-6,-8)(-7,6,-9,-15)(-10,15)(-12,-20,3)(-13,18,4,20)(-14,9,5,-18)(-16,1,7)(-19,12,2,16,10,14)(13,19)
Multiloop annotated with half-edges
12^3_44 annotated with half-edges