$12^{3}_{34}$ - Minimal pinning sets
Pinning sets for 12^3_34
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_34
Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 224
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9785
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{1, 3, 5, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
•
{1, 3, 4, 7, 11}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
•
{1, 2, 3, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.27
6
0
0
18
2.59
7
0
0
46
2.82
8
0
0
65
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
0
221
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,7,0],[0,7,7,1],[1,8,5,5],[1,4,4,9],[2,9,8,7],[2,6,3,3],[4,6,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[4,16,1,5],[5,9,6,8],[15,3,16,4],[1,10,2,9],[6,17,7,20],[7,19,8,20],[14,11,15,12],[2,10,3,11],[17,14,18,13],[18,12,19,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(15,12,-16,-13)(16,3,-5,-4)(4,5,-1,-6)(9,6,-10,-7)(18,7,-19,-8)(8,17,-9,-18)(2,11,-3,-12)(14,19,-15,-20)(20,13,-17,-14)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,6)(-2,-12,15,19,7,-10)(-3,16,12)(-4,-6,9,17,13,-16)(-5,4)(-7,18,-9)(-8,-18)(-11,2)(-13,20,-15)(-14,-20)(-17,8,-19,14)(1,5,3,11)
Multiloop annotated with half-edges
12^3_34 annotated with half-edges