$12^{2}_{117}$ - Minimal pinning sets
Pinning sets for 12^2_117
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_117
Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 320
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: 3.03463
on average over minimal pinning sets: 2.325
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{2, 3, 6, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
•
{1, 2, 4, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
1
8
2.56
6
0
0
34
2.77
7
0
0
71
2.94
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
1
318
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,6,3],[0,2,6,7],[0,5,1,1],[1,4,8,9],[2,7,3,2],[3,6,9,8],[5,7,9,9],[5,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[3,14,4,1],[2,20,3,15],[6,13,7,14],[4,7,5,8],[1,16,2,15],[16,19,17,20],[12,5,13,6],[8,12,9,11],[18,10,19,11],[17,10,18,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,3,-11,-4)(2,5,-3,-6)(7,18,-8,-19)(19,8,-20,-9)(9,6,-10,-7)(4,11,-5,-12)(1,12,-2,-13)(17,20,-18,-15)(14,15,-1,-16)(16,13,-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,-13,16)(-2,-6,9,-20,17,13)(-3,10,6)(-4,-12,1,15,-18,7,-10)(-5,2,12)(-7,-19,-9)(-8,19)(-11,4)(-14,-16)(-15,14,-17)(3,5,11)(8,18,20)
Multiloop annotated with half-edges
12^2_117 annotated with half-edges