$12^{2}_{228}$ - Minimal pinning sets
Pinning sets for 12^2_228 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_228 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, 2, 6, 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, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,7,7],[0,7,7,8],[0,9,5,5],[1,4,4,9],[1,9,8,8],[2,3,3,2],[3,6,6,9],[4,8,6,5]]
PD code (use to draw this multiloop with SnapPy): [[14,7,1,8],[8,3,9,4],[4,13,5,14],[6,11,7,12],[1,15,2,20],[2,19,3,20],[9,16,10,17],[12,5,13,6],[17,10,18,11],[15,18,16,19]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(10,3,-11,-4)(12,5,-13,-6)(19,8,-20,-9)(4,9,-5,-10)(2,11,-3,-12)(7,18,-8,-19)(17,20,-18,-15)(14,15,-1,-16)(16,13,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,-13,16)(-2,-12,-6)(-3,10,-5,12)(-4,-10)(-7,-19,-9,4,-11,2)(-8,19)(-14,-16)(-15,14,-17)(-18,7,1,15)(-20,17,13,5,9)(3,11)(8,18,20)
Multiloop annotated with half-edges
12^2_228 annotated with half-edges