$11^{2}_{54}$ - Minimal pinning sets
Pinning sets for 11^2_54
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_54
Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 168
of which optimal: 3
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.01171
on average over minimal pinning sets: 2.6
on average over optimal pinning sets: 2.6
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{2, 3, 5, 7, 8}
5
[2, 2, 2, 3, 4]
2.60
B (optimal)
•
{3, 5, 6, 8, 10}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
•
{3, 4, 5, 8, 9}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
•
{2, 3, 5, 6, 8, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
•
{3, 4, 5, 7, 8, 10}
6
[2, 2, 2, 4, 4, 4]
3.00
c (minimal)
•
{1, 3, 5, 6, 8, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
d (minimal)
•
{1, 2, 3, 5, 6, 8}
6
[2, 2, 2, 3, 3, 3]
2.50
e (minimal)
•
{1, 2, 3, 5, 8, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.6
6
0
5
18
2.76
7
0
0
52
2.95
8
0
0
53
3.08
9
0
0
28
3.17
10
0
0
8
3.23
11
0
0
1
3.27
Total
3
5
160
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,7,8],[0,8,4,4],[0,3,3,5],[1,4,6,6],[1,5,5,7],[2,6,8,8],[2,7,7,3]]
PD code (use to draw this multiloop with SnapPy): [[12,3,1,4],[4,7,5,8],[8,11,9,12],[2,18,3,13],[1,18,2,17],[6,16,7,17],[5,16,6,15],[10,14,11,15],[9,14,10,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(2,5,-3,-6)(6,9,-7,-10)(8,15,-9,-16)(16,3,-17,-4)(4,17,-5,-18)(18,11,-13,-12)(12,13,-1,-14)(14,7,-15,-8)
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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-7,14)(-2,-6,-10)(-3,16,-9,6)(-4,-18,-12,-14,-8,-16)(-5,2,-11,18)(-13,12)(-15,8)(-17,4)(1,13,11)(3,5,17)(7,9,15)
Multiloop annotated with half-edges
11^2_54 annotated with half-edges