$12^{2}_{299}$ - Minimal pinning sets
Pinning sets for 12^2_299 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_299 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 548
of which optimal: 1
of which minimal: 11
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.18028
on average over minimal pinning sets: 2.92576
on average over optimal pinning sets: 2.75
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 5, 7, 11}
4
[2, 3, 3, 3]
2.75
a (minimal)
• {2, 5, 8, 9, 11}
5
[2, 3, 3, 3, 4]
3.00
b (minimal)
• {2, 5, 8, 10, 11}
5
[2, 3, 3, 3, 5]
3.20
c (minimal)
• {2, 5, 6, 8, 11}
5
[2, 3, 3, 3, 4]
3.00
d (minimal)
• {2, 3, 5, 8, 11}
5
[2, 3, 3, 3, 3]
2.80
e (minimal)
• {2, 4, 5, 8, 11}
5
[2, 3, 3, 3, 4]
3.00
f (minimal)
• {1, 3, 5, 7, 12}
5
[2, 3, 3, 3, 3]
2.80
g (minimal)
• {1, 2, 5, 7, 12}
5
[2, 3, 3, 3, 3]
2.80
h (minimal)
• {1, 3, 5, 8, 11, 12}
6
[2, 3, 3, 3, 3, 3]
2.83
i (minimal)
• {1, 3, 5, 8, 9, 12}
6
[2, 3, 3, 3, 3, 4]
3.00
j (minimal)
• {1, 2, 5, 8, 9, 12}
6
[2, 3, 3, 3, 3, 4]
3.00
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
2.75
5
0
7
8
2.93
6
0
3
60
3.04
7
0
0
132
3.13
8
0
0
160
3.2
9
0
0
116
3.26
10
0
0
49
3.3
11
0
0
11
3.32
12
0
0
1
3.33
Total
1
10
537
Other information about this multiloop Properties
Region degree sequence: [2, 3, 3, 3, 3, 3, 3, 3, 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],[0,6,7,8],[0,9,5,1],[1,4,9,2],[2,9,7,3],[3,6,8,8],[3,7,7,9],[4,8,6,5]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,13,6,12],[3,11,4,12],[16,19,17,20],[1,14,2,13],[6,2,7,3],[15,10,16,11],[18,9,19,10],[17,9,18,8],[14,8,15,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(6,13,-7,-14)(7,16,-8,-17)(17,8,-18,-9)(14,9,-15,-10)(1,10,-2,-11)(11,20,-12,-5)(12,3,-13,-4)(15,18,-16,-19)(2,19,-3,-20)
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,-11,-5)(-2,-20,11)(-3,12,20)(-4,5,-12)(-6,-14,-10,1)(-7,-17,-9,14)(-8,17)(-13,6,4)(-15,-19,2,10)(-16,7,13,3,19)(-18,15,9)(8,16,18)
Multiloop annotated with half-edges
12^2_299 annotated with half-edges