$12^{2}_{320}$ - Minimal pinning sets
Pinning sets for 12^2_320 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_320 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)
• {1, 3, 6, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 4, 8, 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, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,6,6,3],[0,2,7,4],[0,3,8,1],[1,9,6,1],[2,5,9,2],[3,9,8,8],[4,7,7,9],[5,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[10,5,1,6],[6,11,7,20],[9,13,10,14],[4,12,5,13],[1,12,2,11],[7,19,8,20],[14,8,15,9],[3,17,4,18],[2,17,3,16],[18,15,19,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,3,-9,-4)(19,6,-20,-7)(2,17,-3,-18)(7,18,-8,-19)(13,20,-14,-11)(11,10,-12,-1)(1,12,-2,-13)(5,14,-6,-15)(15,4,-16,-5)(16,9,-17,-10)
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,-11)(-2,-18,7,-20,13)(-3,8,18)(-4,15,-6,19,-8)(-5,-15)(-7,-19)(-9,16,4)(-10,11,-14,5,-16)(-12,1)(-17,2,12,10)(3,17,9)(6,14,20)
Multiloop annotated with half-edges
12^2_320 annotated with half-edges