$12^{2}_{247}$ - Minimal pinning sets
Pinning sets for 12^2_247
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_247
Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{1, 2, 6, 8, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
•
{1, 3, 6, 8, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,3,3],[0,2,2,7],[0,7,5,5],[1,4,4,6],[1,5,8,8],[3,9,9,4],[6,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[11,14,12,1],[3,10,4,11],[4,13,5,14],[12,5,13,6],[1,15,2,20],[2,19,3,20],[9,18,10,19],[6,16,7,15],[17,8,18,9],[16,8,17,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,3,-11,-4)(5,18,-6,-19)(19,6,-20,-7)(1,8,-2,-9)(2,11,-3,-12)(9,12,-10,-13)(7,20,-8,-15)(14,15,-1,-16)(16,13,-17,-14)(17,4,-18,-5)
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,-13,16)(-2,-12,9)(-3,10,12)(-4,17,13,-10)(-5,-19,-7,-15,14,-17)(-6,19)(-8,1,15)(-11,2,8,20,6,18,4)(-14,-16)(-18,5)(-20,7)(3,11)
Multiloop annotated with half-edges
12^2_247 annotated with half-edges