$12^{3}_{48}$ - Minimal pinning sets
Pinning sets for 12^3_48
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_48
Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 188
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97456
on average over minimal pinning sets: 2.40667
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{1, 2, 3, 5, 7}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
•
{1, 2, 4, 5, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
•
{1, 2, 4, 5, 7, 9}
6
[2, 2, 2, 2, 3, 5]
2.67
c (minimal)
•
{1, 2, 4, 5, 7, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
d (minimal)
•
{1, 2, 4, 5, 7, 12}
6
[2, 2, 2, 2, 3, 4]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
4
7
2.5
7
0
0
35
2.77
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
4
183
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,3,3],[0,2,2,6],[0,7,8,8],[1,8,9,1],[3,9,9,7],[4,6,9,8],[4,7,5,4],[5,7,6,6]]
PD code (use to draw this multiloop with SnapPy): [[5,8,6,1],[4,14,5,9],[7,13,8,14],[6,13,7,12],[1,15,2,20],[9,3,10,4],[17,11,18,12],[15,18,16,19],[2,19,3,20],[10,16,11,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(2,5,-3,-6)(18,11,-19,-12)(12,3,-13,-4)(4,13,-5,-14)(1,14,-2,-9)(16,19,-17,-20)(7,20,-8,-15)(15,6,-16,-7)(10,17,-11,-18)
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,-9)(-2,-6,15,-8,9)(-3,12,-19,16,6)(-4,-14,1,-10,-18,-12)(-5,2,14)(-7,-15)(-11,18)(-13,4)(-16,-20,7)(-17,10,8,20)(3,5,13)(11,17,19)
Multiloop annotated with half-edges
12^3_48 annotated with half-edges