$12^{2}_{88}$ - Minimal pinning sets
Pinning sets for 12^2_88 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_88 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 320
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03463
on average over minimal pinning sets: 2.325
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 9, 12}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 8, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
1
8
2.56
6
0
0
34
2.77
7
0
0
71
2.94
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
1
318
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,7,7,3],[0,2,8,9],[0,5,5,1],[1,4,4,6],[1,5,9,7],[2,6,8,2],[3,7,9,9],[3,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,10,8,11],[15,5,16,6],[16,19,17,20],[1,9,2,10],[8,2,9,3],[11,3,12,4],[4,14,5,15],[18,13,19,14],[17,13,18,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(11,2,-12,-3)(18,5,-19,-6)(4,19,-5,-20)(17,20,-18,-7)(8,13,-9,-14)(14,9,-15,-10)(1,10,-2,-11)(12,15,-13,-16)(3,16,-4,-17)
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,-3,-17,-7)(-2,11)(-4,-20,17)(-5,18,20)(-6,7,-18)(-8,-14,-10,1)(-9,14)(-12,-16,3)(-13,8,6,-19,4,16)(-15,12,2,10)(5,19)(9,13,15)
Multiloop annotated with half-edges
12^2_88 annotated with half-edges