$12^{2}_{54}$ - Minimal pinning sets
Pinning sets for 12^2_54 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_54 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, 3, 6, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 4, 5, 6, 9}
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: [[1,2,3,3],[0,4,4,5],[0,6,7,3],[0,2,4,0],[1,3,5,1],[1,4,8,6],[2,5,9,7],[2,6,9,8],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[16,20,1,17],[17,8,18,7],[15,2,16,3],[19,1,20,2],[8,19,9,18],[9,6,10,7],[3,10,4,11],[11,14,12,15],[12,5,13,6],[4,13,5,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,16,-14,-1)(10,3,-11,-4)(7,4,-8,-5)(5,20,-6,-17)(1,8,-2,-9)(2,11,-3,-12)(9,12,-10,-13)(18,15,-19,-16)(17,6,-18,-7)(14,19,-15,-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,-9,-13)(-2,-12,9)(-3,10,12)(-4,7,-18,-16,13,-10)(-5,-17,-7)(-6,17)(-8,1,-14,-20,5)(-11,2,8,4)(-15,18,6,20)(-19,14,16)(3,11)(15,19)
Multiloop annotated with half-edges
12^2_54 annotated with half-edges