$12^{2}_{183}$ - Minimal pinning sets
Pinning sets for 12^2_183 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_183 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 136
of which optimal: 3
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97614
on average over minimal pinning sets: 2.39286
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, 2, 5, 6, 11, 12}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 5, 6, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 2, 4, 5, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
a (minimal)
• {1, 2, 4, 5, 7, 11, 12}
7
[2, 2, 2, 2, 3, 3, 4]
2.57
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
3
0
0
2.33
7
0
1
16
2.64
8
0
0
38
2.88
9
0
0
43
3.05
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
1
132
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,4,5],[0,6,6,0],[1,7,2,1],[2,7,8,9],[3,9,7,3],[4,6,8,5],[5,7,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,10,14,9],[11,8,12,9],[19,1,20,2],[10,15,11,14],[4,7,5,8],[2,18,3,19],[15,3,16,4],[16,6,17,7],[5,17,6,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,1,-19,-2)(19,4,-20,-5)(2,5,-3,-6)(9,6,-10,-7)(7,16,-8,-17)(14,11,-15,-12)(3,20,-4,-13)(12,13,-1,-14)(10,15,-11,-16)(17,8,-18,-9)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,8,16,-11,14)(-2,-6,9,-18)(-3,-13,12,-15,10,6)(-4,19,1,13)(-5,2,-19)(-7,-17,-9)(-8,17)(-10,-16,7)(-12,-14)(-20,3,5)(4,20)(11,15)
Multiloop annotated with half-edges
12^2_183 annotated with half-edges