$12^{2}_{225}$ - Minimal pinning sets
Pinning sets for 12^2_225 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_225 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 256
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: 2.96564
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, 3, 5, 9}
4
[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
4
1
0
0
2.0
5
0
0
8
2.4
6
0
0
28
2.67
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
0
255
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,6],[0,7,7,4],[1,3,8,9],[2,9,6,6],[2,5,5,7],[3,6,8,3],[4,7,9,9],[4,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,13,16,14],[16,19,17,20],[1,8,2,9],[9,12,10,13],[18,4,19,5],[17,4,18,3],[7,2,8,3],[11,6,12,7],[10,6,11,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,15,-1,-16)(16,1,-17,-2)(2,7,-3,-8)(8,3,-9,-4)(12,5,-13,-6)(6,9,-7,-10)(19,10,-20,-11)(4,13,-5,-14)(20,17,-15,-18)(11,18,-12,-19)
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)(-2,-8,-4,-14,-16)(-3,8)(-5,12,18,-15,14)(-6,-10,19,-12)(-7,2,-17,20,10)(-9,6,-13,4)(-11,-19)(-18,11,-20)(1,15,17)(3,7,9)(5,13)
Multiloop annotated with half-edges
12^2_225 annotated with half-edges