$12^{2}_{149}$ - Minimal pinning sets
Pinning sets for 12^2_149 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_149 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 144
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97193
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 7, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 3, 4, 7, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 2, 3, 4, 7, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
D (optimal)
• {1, 2, 3, 4, 7, 8}
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
6
4
0
0
2.33
7
0
0
20
2.66
8
0
0
41
2.89
9
0
0
44
3.06
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
4
0
140
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,3,0],[0,2,7,1],[1,7,8,5],[1,4,8,8],[2,8,9,9],[3,9,9,4],[4,6,5,5],[6,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,8,12,7],[19,9,20,10],[1,9,2,8],[12,16,13,15],[6,14,7,15],[18,4,19,5],[2,17,3,16],[13,5,14,6],[3,17,4,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,10,-12,-1)(2,19,-3,-20)(3,6,-4,-7)(17,4,-18,-5)(16,9,-17,-10)(1,12,-2,-13)(13,20,-14,-11)(14,7,-15,-8)(8,15,-9,-16)(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,-13,-11)(-2,-20,13)(-3,-7,14,20)(-4,17,9,15,7)(-5,-19,2,12,10,-17)(-6,3,19)(-8,-16,-10,11,-14)(-9,16)(-12,1)(-15,8)(-18,5)(4,6,18)
Multiloop annotated with half-edges
12^2_149 annotated with half-edges