$12^{4}_{3}$ - Minimal pinning sets
Pinning sets for 12^4_3 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^4_3 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)
• {2, 4, 6, 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, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,6,7,3],[0,2,4,0],[1,3,5,1],[1,4,8,9],[2,9,7,7],[2,6,6,8],[5,7,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[6,10,1,7],[7,11,8,16],[5,2,6,3],[9,1,10,2],[11,9,12,8],[12,15,13,16],[3,17,4,20],[4,19,5,20],[14,18,15,19],[13,18,14,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,3,-17,-4)(1,4,-2,-5)(19,14,-20,-15)(2,17,-3,-18)(15,12,-16,-13)(13,18,-14,-19)(9,16,-10,-11)(11,10,-12,-7)(6,7,-1,-8)(8,5,-9,-6)
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,-5,8)(-2,-18,13,-16,9,5)(-3,20,14,18)(-4,1,7,-12,15,-20)(-6,-8)(-7,6,-9,-11)(-10,11)(-13,-19,-15)(-14,19)(-17,2,4)(3,17)(10,16,12)
Multiloop annotated with half-edges
12^4_3 annotated with half-edges