$12^{2}_{13}$ - Minimal pinning sets
Pinning sets for 12^2_13 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_13 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 484
of which optimal: 3
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04857
on average over minimal pinning sets: 2.6
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, 3, 7}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 3, 7, 12}
4
[2, 2, 2, 4]
2.50
C (optimal)
• {1, 3, 7, 8}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 5, 7, 11}
5
[2, 2, 2, 4, 4]
2.80
b (minimal)
• {1, 3, 7, 9, 11}
5
[2, 2, 2, 4, 4]
2.80
c (minimal)
• {1, 3, 5, 6, 7}
5
[2, 2, 2, 4, 4]
2.80
d (minimal)
• {1, 3, 6, 7, 9}
5
[2, 2, 2, 4, 4]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
3
0
0
2.33
5
0
4
21
2.67
6
0
0
76
2.88
7
0
0
124
3.01
8
0
0
126
3.11
9
0
0
84
3.19
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
4
477
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],[0,4,4,5],[0,5,6,7],[0,8,6,5],[0,5,1,1],[1,4,3,2],[2,3,9,9],[2,9,8,8],[3,7,7,9],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[3,12,4,1],[2,20,3,13],[15,11,16,12],[4,18,5,19],[1,14,2,13],[14,19,15,20],[5,10,6,11],[16,9,17,8],[17,7,18,8],[9,6,10,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,4,-8,-5)(18,5,-19,-6)(6,17,-7,-18)(3,8,-4,-9)(16,9,-17,-10)(1,10,-2,-11)(2,19,-3,-20)(15,20,-16,-13)(12,13,-1,-14)(14,11,-15,-12)
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,-11,14)(-2,-20,15,11)(-3,-9,16,20)(-4,7,17,9)(-5,18,-7)(-6,-18)(-8,3,19,5)(-10,1,13,-16)(-12,-14)(-13,12,-15)(-17,6,-19,2,10)(4,8)
Multiloop annotated with half-edges
12^2_13 annotated with half-edges