$12^{3}_{28}$ - Minimal pinning sets
Pinning sets for 12^3_28
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_28
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, 3, 6, 9}
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, 3, 4, 5, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,6,7,7],[0,8,8,0],[1,8,5,1],[1,4,9,6],[2,5,9,9],[2,9,8,2],[3,7,4,3],[5,7,6,6]]
PD code (use to draw this multiloop with SnapPy): [[5,10,6,1],[4,16,5,11],[9,20,10,17],[6,2,7,1],[11,3,12,4],[12,15,13,16],[17,13,18,14],[19,8,20,9],[2,8,3,7],[14,18,15,19]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,10,-12,-1)(4,9,-5,-10)(20,5,-17,-6)(2,7,-3,-8)(8,3,-9,-4)(18,13,-19,-14)(12,15,-13,-16)(1,16,-2,-11)(6,17,-7,-18)(14,19,-15,-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,-11)(-2,-8,-4,-10,11)(-3,8)(-5,20,-15,12,10)(-6,-18,-14,-20)(-7,2,16,-13,18)(-9,4)(-12,-16,1)(-17,6)(-19,14)(3,7,17,5,9)(13,15,19)
Multiloop annotated with half-edges
12^3_28 annotated with half-edges