$12^{4}_{5}$ - Minimal pinning sets
Pinning sets for 12^4_5 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^4_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, 7, 12}
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],[0,4,5,0],[1,6,7,1],[2,7,7,8],[2,8,6,6],[3,5,5,9],[3,9,4,4],[4,9,9,5],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[6,12,1,7],[7,5,8,6],[11,1,12,2],[4,8,5,9],[2,13,3,16],[10,20,11,17],[9,20,10,19],[3,13,4,14],[15,17,16,18],[18,14,19,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-7,-2)(10,3,-11,-4)(6,7,-1,-8)(2,11,-3,-12)(17,4,-18,-5)(5,18,-6,-19)(16,19,-13,-20)(13,8,-14,-9)(9,14,-10,-15)(20,15,-17,-16)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-3,10,14,8)(-2,-12)(-4,17,15,-10)(-5,-19,16,-17)(-6,-8,13,19)(-7,6,18,4,-11,2)(-9,-15,20,-13)(-14,9)(-16,-20)(-18,5)(1,7)(3,11)
Multiloop annotated with half-edges
12^4_5 annotated with half-edges