$12^{2}_{24}$ - Minimal pinning sets
Pinning sets for 12^2_24 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_24 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, 4, 7, 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, 3, 4, 4, 4, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,2],[0,3,3,0],[0,4,5,0],[1,6,7,1],[2,8,8,9],[2,9,6,6],[3,5,5,9],[3,9,8,8],[4,7,7,4],[4,7,6,5]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,13,16,14],[19,1,20,2],[12,16,13,17],[2,10,3,9],[6,18,7,19],[17,7,18,8],[11,4,12,5],[10,4,11,3],[5,8,6,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,8,-2,-9)(9,2,-10,-3)(20,5,-15,-6)(3,6,-4,-7)(7,14,-8,-1)(18,11,-19,-12)(16,13,-17,-14)(10,19,-11,-20)(4,15,-5,-16)(12,17,-13,-18)
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,-3,-7)(-2,9)(-4,-16,-14,7)(-5,20,-11,18,-13,16)(-6,3,-10,-20)(-8,1)(-12,-18)(-15,4,6)(-17,12,-19,10,2,8,14)(5,15)(11,19)(13,17)
Multiloop annotated with half-edges
12^2_24 annotated with half-edges