$12^{3}_{22}$ - Minimal pinning sets
Pinning sets for 12^3_22 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_22 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, 7, 11}
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, 3, 3, 7, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,3,3],[0,2,2,6],[0,5,1,1],[1,4,7,8],[2,9,9,3],[5,9,8,8],[5,7,7,9],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[3,6,4,1],[2,12,3,7],[5,20,6,13],[4,20,5,19],[1,8,2,7],[8,11,9,12],[13,18,14,19],[10,16,11,17],[9,16,10,15],[17,14,18,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,4,-2,-5)(9,12,-10,-7)(6,7,-1,-8)(8,5,-9,-6)(17,10,-18,-11)(11,18,-12,-19)(15,20,-16,-13)(13,2,-14,-3)(3,14,-4,-15)(19,16,-20,-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,-5,8)(-2,13,-16,19,-12,9,5)(-3,-15,-13)(-4,1,7,-10,17,-20,15)(-6,-8)(-7,6,-9)(-11,-19,-17)(-14,3)(-18,11)(2,4,14)(10,12,18)(16,20)
Multiloop annotated with half-edges
12^3_22 annotated with half-edges