$12^{2}_{307}$ - Minimal pinning sets
Pinning sets for 12^2_307 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_307 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, 7, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 4, 7, 11}
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, 4, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,6],[0,6,6,7],[0,7,8,1],[1,8,8,2],[2,9,3,3],[3,9,9,4],[4,9,5,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[5,16,6,1],[4,9,5,10],[15,8,16,9],[6,17,7,20],[1,11,2,10],[14,3,15,4],[7,17,8,18],[19,11,20,12],[2,13,3,14],[18,13,19,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(13,6,-14,-7)(2,7,-3,-8)(8,15,-9,-16)(20,9,-17,-10)(10,19,-11,-20)(16,11,-1,-12)(3,14,-4,-15)(17,4,-18,-5)(5,18,-6,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12)(-2,-8,-16,-12)(-3,-15,8)(-4,17,9,15)(-5,-19,10,-17)(-6,13,1,11,19)(-7,2,-13)(-9,20,-11,16)(-10,-20)(-14,3,7)(-18,5)(4,14,6,18)
Multiloop annotated with half-edges
12^2_307 annotated with half-edges