$12^{4}_{6}$ - Minimal pinning sets
Pinning sets for 12^4_6 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^4_6 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 128
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90623
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 7, 12}
5
[2, 2, 2, 2, 2]
2.00
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
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,7,7],[0,7,8,8],[0,8,8,5],[1,4,9,1],[1,9,9,2],[2,9,3,2],[3,4,4,3],[5,7,6,6]]
PD code (use to draw this multiloop with SnapPy): [[6,12,1,7],[7,13,8,16],[5,20,6,17],[11,3,12,4],[1,10,2,9],[13,9,14,8],[15,17,16,18],[19,4,20,5],[2,10,3,11],[14,19,15,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(17,2,-18,-3)(11,14,-12,-15)(12,5,-7,-6)(6,7,-1,-8)(4,9,-5,-10)(13,10,-14,-11)(20,15,-17,-16)(3,18,-4,-19)(16,19,-13,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,17,15,-12,-6,-8)(-3,-19,16,-17)(-4,-10,13,19)(-5,12,14,10)(-7,6)(-9,4,18,2)(-11,-15,20,-13)(-14,11)(-16,-20)(-18,3)(1,7,5,9)
Multiloop annotated with half-edges
12^4_6 annotated with half-edges