$12^{2}_{43}$ - Minimal pinning sets
Pinning sets for 12^2_43 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_43 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 160
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: 2.97043
on average over minimal pinning sets: 2.26667
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, 3, 4, 7}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 5, 7, 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
1
7
2.5
7
0
0
26
2.74
8
0
0
45
2.92
9
0
0
45
3.07
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
158
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,7,7],[0,8,9,9],[0,5,5,1],[1,4,4,6],[1,5,9,2],[2,8,8,2],[3,7,7,9],[3,8,6,3]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,16,14,17],[4,11,5,12],[7,19,8,20],[1,15,2,16],[14,2,15,3],[17,3,18,4],[10,5,11,6],[6,9,7,10],[18,8,19,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,12,-14,-1)(16,1,-17,-2)(8,3,-9,-4)(17,6,-18,-7)(4,7,-5,-8)(2,9,-3,-10)(11,14,-12,-15)(20,15,-13,-16)(5,18,-6,-19)(10,19,-11,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-13)(-2,-10,-20,-16)(-3,8,-5,-19,10)(-4,-8)(-6,17,1,-14,11,19)(-7,4,-9,2,-17)(-11,-15,20)(-12,13,15)(-18,5,7)(3,9)(6,18)(12,14)
Multiloop annotated with half-edges
12^2_43 annotated with half-edges