$12^{3}_{42}$ - Minimal pinning sets
Pinning sets for 12^3_42
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_42
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, 3, 5, 8, 9, 11}
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, 3, 3, 5, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,3,4],[0,5,6,0],[0,7,1,1],[1,8,8,5],[2,4,6,6],[2,5,5,7],[3,6,9,9],[4,9,9,4],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[3,8,4,1],[2,14,3,9],[7,4,8,5],[1,10,2,9],[13,20,14,15],[5,20,6,19],[6,18,7,19],[10,18,11,17],[15,12,16,13],[11,16,12,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,2,-6,-3)(1,6,-2,-7)(18,13,-19,-14)(8,9,-1,-10)(10,7,-11,-8)(20,11,-15,-12)(4,15,-5,-16)(16,3,-17,-4)(14,17,-9,-18)(12,19,-13,-20)
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,-7,10)(-2,5,15,11,7)(-3,16,-5)(-4,-16)(-6,1,9,17,3)(-8,-10)(-9,8,-11,20,-13,18)(-12,-20)(-14,-18)(-15,4,-17,14,-19,12)(2,6)(13,19)
Multiloop annotated with half-edges
12^3_42 annotated with half-edges