$12^{2}_{4}$ - Minimal pinning sets
Pinning sets for 12^2_4 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_4 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, 8, 9, 11}
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, 3, 4, 4, 7, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,5,4,3],[0,2,1,0],[1,2,6,6],[1,7,7,2],[4,8,8,4],[5,9,9,5],[6,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,19,12,18],[9,2,10,3],[19,1,20,2],[12,9,13,8],[17,3,18,4],[13,7,14,8],[4,16,5,17],[6,14,7,15],[15,5,16,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-11,-2)(18,3,-19,-4)(4,17,-5,-18)(14,7,-15,-8)(5,8,-6,-9)(12,9,-13,-10)(10,11,-1,-12)(13,16,-14,-17)(6,15,-7,-16)(2,19,-3,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,-3,18,-5,-9,12)(-2,-20)(-4,-18)(-6,-16,13,9)(-7,14,16)(-8,5,17,-14)(-10,-12)(-11,10,-13,-17,4,-19,2)(-15,6,8)(1,11)(3,19)(7,15)
Multiloop annotated with half-edges
12^2_4 annotated with half-edges