$12^{2}_{28}$ - Minimal pinning sets
Pinning sets for 12^2_28 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_28 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 64
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: 2.85421
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, 3, 4, 8, 9}
6
[2, 2, 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
6
1
0
0
2.0
7
0
0
6
2.38
8
0
0
15
2.67
9
0
0
20
2.89
10
0
0
15
3.07
11
0
0
6
3.21
12
0
0
1
3.33
Total
1
0
63
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,1,2,2],[0,3,3,0],[0,4,4,0],[1,5,6,1],[2,6,5,2],[3,4,7,8],[3,8,7,4],[5,6,9,9],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,9,12,10],[19,1,20,2],[8,12,9,13],[2,18,3,19],[13,3,14,4],[17,7,18,8],[14,7,15,6],[4,16,5,17],[15,5,16,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-17,-2)(18,3,-19,-4)(4,17,-5,-18)(20,5,-11,-6)(14,7,-15,-8)(12,9,-13,-10)(6,15,-7,-16)(2,19,-3,-20)(10,11,-1,-12)(8,13,-9,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-7,14,-9,12)(-2,-20,-6,-16)(-3,18,-5,20)(-4,-18)(-8,-14)(-10,-12)(-11,10,-13,8,-15,6)(-17,4,-19,2)(1,11,5,17)(3,19)(7,15)(9,13)
Multiloop annotated with half-edges
12^2_28 annotated with half-edges