$12^{2}_{221}$ - Minimal pinning sets
Pinning sets for 12^2_221
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_221
Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 140
of which optimal: 3
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97615
on average over minimal pinning sets: 2.42857
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{1, 3, 4, 7, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
•
{1, 2, 3, 7, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
•
{1, 2, 3, 6, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
a (minimal)
•
{1, 3, 4, 5, 6, 9, 11}
7
[2, 2, 2, 2, 3, 3, 4]
2.57
b (minimal)
•
{1, 3, 4, 6, 8, 9, 11}
7
[2, 2, 2, 2, 3, 3, 4]
2.57
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
3
0
0
2.33
7
0
2
16
2.63
8
0
0
40
2.88
9
0
0
44
3.06
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
2
135
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,7,4],[0,3,7,5],[1,4,2,1],[2,8,8,3],[3,9,9,4],[6,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,8,12,7],[9,6,10,7],[19,14,20,15],[1,14,2,13],[8,13,9,12],[5,15,6,16],[18,2,19,3],[16,4,17,5],[3,17,4,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,10,-12,-1)(19,2,-20,-3)(7,4,-8,-5)(5,14,-6,-15)(15,6,-16,-7)(16,9,-17,-10)(12,17,-13,-18)(3,18,-4,-19)(1,20,-2,-11)(8,13,-9,-14)
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)(-2,19,-4,7,-16,-10,11)(-3,-19)(-5,-15,-7)(-6,15)(-8,-14,5)(-9,16,6,14)(-12,-18,3,-20,1)(-13,8,4,18)(-17,12,10)(2,20)(9,13,17)
Multiloop annotated with half-edges
12^2_221 annotated with half-edges