$12^{3}_{33}$ - Minimal pinning sets
Pinning sets for 12^3_33
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_33
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, 5, 7}
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, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,5,5,3],[0,2,6,7],[0,8,9,1],[1,2,2,1],[3,9,7,7],[3,6,6,8],[4,7,9,9],[4,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,5,10,6],[11,7,12,8],[12,15,13,16],[1,4,2,5],[10,7,11,6],[14,20,15,17],[13,20,14,19],[3,18,4,19],[2,18,3,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(13,2,-14,-3)(3,14,-4,-15)(15,4,-16,-5)(16,19,-17,-20)(5,20,-6,-13)(12,7,-9,-8)(8,9,-1,-10)(10,17,-11,-18)(18,11,-19,-12)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,20,-17,10)(-2,13,-6)(-3,-15,-5,-13)(-4,15)(-7,12,-19,16,4,14,2)(-8,-10,-18,-12)(-9,8)(-11,18)(-14,3)(-16,-20,5)(1,9,7)(11,17,19)
Multiloop annotated with half-edges
12^3_33 annotated with half-edges