$12^{2}_{169}$ - Minimal pinning sets
Pinning sets for 12^2_169
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_169
Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
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: 2.96906
on average over minimal pinning sets: 2.2
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, 3, 4, 5, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
•
{1, 2, 4, 5, 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
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
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
2
0
190
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,7],[1,7,7,5],[2,4,8,6],[3,5,9,9],[3,8,4,4],[5,7,9,9],[6,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[14,7,1,8],[8,13,9,14],[9,6,10,7],[1,10,2,11],[12,20,13,15],[5,19,6,20],[2,19,3,18],[11,16,12,15],[16,4,17,5],[3,17,4,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,14,-16,-1)(5,2,-6,-3)(10,3,-11,-4)(11,6,-12,-7)(4,9,-5,-10)(13,16,-14,-17)(17,12,-18,-13)(1,18,-2,-19)(8,19,-9,-20)(20,7,-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)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-19,8,-15)(-2,5,9,19)(-3,10,-5)(-4,-10)(-6,11,3)(-7,20,-9,4,-11)(-8,-20)(-12,17,-14,15,7)(-13,-17)(-16,13,-18,1)(2,18,12,6)(14,16)
Multiloop annotated with half-edges
12^2_169 annotated with half-edges