$12^{3}_{58}$ - Minimal pinning sets
Pinning sets for 12^3_58
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_58
Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 128
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: 2.90623
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, 4, 7, 11}
5
[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
5
1
0
0
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 7]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,7],[0,7,8,8],[0,8,8,5],[1,4,6,1],[2,5,9,9],[2,9,9,3],[3,4,4,3],[6,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[8,14,1,9],[9,15,10,20],[7,19,8,20],[13,3,14,4],[1,12,2,11],[15,11,16,10],[16,6,17,7],[18,4,19,5],[2,12,3,13],[5,17,6,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(15,2,-16,-3)(19,4,-20,-5)(13,18,-14,-19)(14,7,-9,-8)(8,9,-1,-10)(6,11,-7,-12)(3,20,-4,-15)(5,16,-6,-17)(17,12,-18,-13)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,15,-4,19,-14,-8,-10)(-3,-15)(-5,-17,-13,-19)(-6,-12,17)(-7,14,18,12)(-9,8)(-11,6,16,2)(-16,5,-20,3)(-18,13)(1,9,7,11)(4,20)
Multiloop annotated with half-edges
12^3_58 annotated with half-edges