$12^{3}_{17}$ - Minimal pinning sets
Pinning sets for 12^3_17
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_17
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, 3, 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, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,5,6,7],[0,7,4,0],[1,3,5,1],[1,4,8,2],[2,8,9,7],[2,6,9,3],[5,9,9,6],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,12,1,9],[9,13,10,20],[15,7,16,8],[11,1,12,2],[13,11,14,10],[14,19,15,20],[6,3,7,4],[16,3,17,2],[18,4,19,5],[5,17,6,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,9,-1,-10)(4,1,-5,-2)(11,2,-12,-3)(16,5,-9,-6)(20,7,-17,-8)(19,14,-20,-15)(6,17,-7,-18)(3,10,-4,-11)(15,12,-16,-13)(13,18,-14,-19)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,10)(-2,11,-4)(-3,-11)(-5,16,12,2)(-6,-18,13,-16)(-7,20,14,18)(-8,-10,3,-12,15,-20)(-9,8,-17,6)(-13,-19,-15)(-14,19)(1,9,5)(7,17)
Multiloop annotated with half-edges
12^3_17 annotated with half-edges