$12^{2}_{212}$ - Minimal pinning sets
Pinning sets for 12^2_212 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_212 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, 4, 5, 7, 11}
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],[0,5,5,2],[0,1,6,3],[0,2,6,6],[0,7,8,8],[1,8,9,1],[2,7,3,3],[4,6,9,9],[4,9,5,4],[5,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[5,16,6,1],[11,4,12,5],[12,15,13,16],[6,13,7,14],[1,17,2,20],[3,10,4,11],[14,7,15,8],[17,8,18,9],[2,19,3,20],[9,18,10,19]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,16,-8,-1)(10,3,-11,-4)(17,4,-18,-5)(15,6,-16,-7)(2,9,-3,-10)(8,11,-9,-12)(1,12,-2,-13)(5,18,-6,-19)(14,19,-15,-20)(20,13,-17,-14)
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,-13,20,-15,-7)(-2,-10,-4,17,13)(-3,10)(-5,-19,14,-17)(-6,15,19)(-8,-12,1)(-9,2,12)(-11,8,16,6,18,4)(-14,-20)(-16,7)(-18,5)(3,9,11)
Multiloop annotated with half-edges
12^2_212 annotated with half-edges