$12^{3}_{47}$ - Minimal pinning sets
Pinning sets for 12^3_47 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_47 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 256
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: 2.96564
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, 5, 9}
4
[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
4
1
0
0
2.0
5
0
0
8
2.4
6
0
0
28
2.67
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
0
255
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,6],[0,6,7,8],[1,8,8,9],[2,9,6,6],[2,5,5,3],[3,9,9,8],[3,7,4,4],[4,7,7,5]]
PD code (use to draw this multiloop with SnapPy): [[10,16,1,11],[11,9,12,10],[12,15,13,16],[1,6,2,7],[8,20,9,17],[4,14,5,15],[13,5,14,6],[2,19,3,18],[7,18,8,17],[3,19,4,20]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,2,-16,-3)(8,3,-9,-4)(19,6,-20,-7)(1,16,-2,-17)(20,17,-15,-18)(7,18,-8,-19)(14,9,-11,-10)(10,11,-1,-12)(5,12,-6,-13)(13,4,-14,-5)
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,-17,20,6,12)(-2,15,17)(-3,8,18,-15)(-4,13,-6,19,-8)(-5,-13)(-7,-19)(-9,14,4)(-10,-12,5,-14)(-11,10)(-16,1,11,9,3)(-18,7,-20)(2,16)
Multiloop annotated with half-edges
12^3_47 annotated with half-edges