$12^{2}_{5}$ - Minimal pinning sets
Pinning sets for 12^2_5 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_5 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, 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, 5, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,6,6,3],[0,2,1,0],[1,7,8,8],[1,9,9,7],[2,7,7,2],[4,6,6,5],[4,9,9,4],[5,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,19,14,18],[11,2,12,3],[19,1,20,2],[14,9,15,8],[17,5,18,6],[3,10,4,11],[9,4,10,5],[15,7,16,8],[6,16,7,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(13,2,-14,-3)(19,4,-20,-5)(5,18,-6,-19)(12,7,-1,-8)(15,10,-16,-11)(8,11,-9,-12)(3,20,-4,-13)(14,17,-15,-18)(9,16,-10,-17)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,18,-15,-11,8)(-2,13,-4,19,-6)(-3,-13)(-5,-19)(-7,12,-9,-17,14,2)(-8,-12)(-10,15,17)(-14,-18,5,-20,3)(-16,9,11)(1,7)(4,20)(10,16)
Multiloop annotated with half-edges
12^2_5 annotated with half-edges