$12^{2}_{102}$ - Minimal pinning sets
Pinning sets for 12^2_102
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_102
Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 160
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97043
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{1, 4, 5, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
•
{1, 3, 4, 6, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
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
2.2
6
0
1
7
2.5
7
0
0
26
2.74
8
0
0
45
2.92
9
0
0
45
3.07
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
158
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,6,7,3],[0,2,7,7],[0,6,8,1],[1,9,9,1],[2,8,4,7],[2,6,3,3],[4,6,9,9],[5,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,6,14,5],[8,11,9,12],[9,19,10,20],[1,7,2,6],[14,4,15,5],[17,7,18,8],[18,10,19,11],[2,17,3,16],[3,15,4,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (19,2,-20,-3)(3,18,-4,-19)(4,11,-5,-12)(14,5,-15,-6)(15,8,-16,-9)(6,9,-7,-10)(1,20,-2,-13)(13,12,-14,-1)(7,16,-8,-17)(10,17,-11,-18)
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,-13)(-2,19,-4,-12,13)(-3,-19)(-5,14,12)(-6,-10,-18,3,-20,1,-14)(-7,-17,10)(-8,15,5,11,17)(-9,6,-15)(-11,4,18)(-16,7,9)(2,20)(8,16)
Multiloop annotated with half-edges
12^2_102 annotated with half-edges