$12^{4}_{11}$ - Minimal pinning sets
Pinning sets for 12^4_11 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^4_11 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 664
of which optimal: 2
of which minimal: 16
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.25459
on average over minimal pinning sets: 3.125
on average over optimal pinning sets: 3.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 9}
4
[3, 3, 3, 3]
3.00
B (optimal)
• {2, 3, 7, 11}
4
[3, 3, 3, 3]
3.00
a (minimal)
• {1, 4, 5, 7, 10, 11}
6
[3, 3, 3, 3, 3, 4]
3.17
b (minimal)
• {1, 4, 6, 7, 10, 11}
6
[3, 3, 3, 3, 4, 4]
3.33
c (minimal)
• {1, 4, 6, 7, 9, 11}
6
[3, 3, 3, 3, 3, 4]
3.17
d (minimal)
• {1, 3, 4, 7, 10, 11}
6
[3, 3, 3, 3, 3, 4]
3.17
e (minimal)
• {1, 3, 4, 7, 9, 11}
6
[3, 3, 3, 3, 3, 3]
3.00
f (minimal)
• {2, 3, 5, 7, 9, 12}
6
[3, 3, 3, 3, 3, 4]
3.17
g (minimal)
• {2, 3, 5, 8, 9, 12}
6
[3, 3, 3, 3, 4, 4]
3.33
h (minimal)
• {2, 3, 5, 8, 9, 11}
6
[3, 3, 3, 3, 3, 4]
3.17
i (minimal)
• {2, 3, 4, 5, 9, 12}
6
[3, 3, 3, 3, 3, 4]
3.17
j (minimal)
• {2, 3, 4, 5, 9, 11}
6
[3, 3, 3, 3, 3, 3]
3.00
k (minimal)
• {1, 2, 3, 5, 7, 9}
6
[3, 3, 3, 3, 3, 3]
3.00
l (minimal)
• {1, 2, 3, 5, 8, 9}
6
[3, 3, 3, 3, 3, 4]
3.17
m (minimal)
• {1, 2, 4, 5, 7, 11}
6
[3, 3, 3, 3, 3, 3]
3.00
n (minimal)
• {1, 2, 4, 6, 7, 11}
6
[3, 3, 3, 3, 3, 4]
3.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
3.0
5
0
0
16
3.1
6
0
14
56
3.16
7
0
0
160
3.22
8
0
0
201
3.27
9
0
0
144
3.3
10
0
0
58
3.32
11
0
0
12
3.33
12
0
0
1
3.33
Total
2
14
648
Other information about this multiloop Properties
Region degree sequence: [3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4]
Minimal region degree: 3
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,6,3],[0,2,7,4],[0,3,8,1],[1,8,9,6],[2,5,9,7],[3,6,9,8],[4,7,9,5],[5,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[6,12,1,7],[7,13,8,16],[5,15,6,16],[11,14,12,15],[1,14,2,13],[8,17,9,20],[4,19,5,20],[10,18,11,19],[2,18,3,17],[9,3,10,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(4,11,-5,-12)(16,17,-11,-18)(3,18,-4,-19)(19,12,-20,-13)(20,5,-17,-6)(10,13,-7,-14)(7,6,-8,-1)(15,8,-16,-9)(2,9,-3,-10)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-7)(-2,-10,-14)(-3,-19,-13,10)(-4,-12,19)(-5,20,12)(-6,7,13,-20)(-8,15,1)(-9,2,-15)(-11,4,18)(-16,-18,3,9)(-17,16,8,6)(5,11,17)
Multiloop annotated with half-edges
12^4_11 annotated with half-edges