$12^{3}_{84}$ - Minimal pinning sets
Pinning sets for 12^3_84 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_84 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 176
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97631
on average over minimal pinning sets: 2.34444
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, 5, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {2, 3, 5, 6, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
b (minimal)
• {2, 4, 5, 6, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
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
2
7
2.5
7
0
0
30
2.75
8
0
0
51
2.94
9
0
0
49
3.08
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
2
173
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,2],[0,1,5,6],[0,6,4,0],[1,3,7,7],[1,8,8,2],[2,8,9,3],[4,9,9,4],[5,9,6,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[6,14,1,7],[7,15,8,20],[5,19,6,20],[13,1,14,2],[15,13,16,12],[8,4,9,5],[18,2,19,3],[16,11,17,12],[3,9,4,10],[10,17,11,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(13,2,-14,-3)(18,5,-19,-6)(1,14,-2,-15)(11,16,-12,-17)(17,10,-18,-11)(4,19,-5,-20)(15,12,-16,-7)(3,8,-4,-9)(9,20,-10,-13)
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,-15,-7)(-2,13,-10,17,-12,15)(-3,-9,-13)(-4,-20,9)(-5,18,10,20)(-6,7,-16,11,-18)(-8,3,-14,1)(-11,-17)(-19,4,8,6)(2,14)(5,19)(12,16)
Multiloop annotated with half-edges
12^3_84 annotated with half-edges